It’s a time-honored tradition for politicians to deny any interest in the vice presidency. But this year, with the possibility of Donald J. Trump as the Republican nominee, they really mean it.
“Never,” said Chris Schrimpf, a spokesman for Gov. John Kasich of Ohio, who is still running against Mr. Trump. “No chance.”
“Hahahahahahahahaha,” wrote Sally Bradshaw, a senior adviser to Jeb Bush, when asked if he would consider it.
“Scott Walker has a visceral negative reaction to Trump’s character,” said Ed Goeas, a longtime adviser to the Wisconsin governor.
Or, as Senator Lindsey Graham put it, “That’s like buying a ticket on the Titanic.”
A remarkable range of leading Republicans, including Gov. Nikki Haley of South Carolina and Senator Jeff Flake of Arizona, have been emphatic publicly or with their advisers and allies that they do not want to be considered as Mr. Trump’s running mate. The recoiling amounts to a rare rebuke for a front-runner: Politicians usually signal that they are not interested politely through back channels, or submit to the selection process, if only to burnish their national profiles.
But Mr. Trump has a singular track record of picking fights with obvious potential running mates like Senator Marco Rubio of Florida, who has indicated a lack of interest in the vice presidency generally and has yet to reconcile with Mr. Trump publicly. Ms. Haley and another potential pick, Gov. Susana Martinez of New Mexico, have sharply criticized Mr. Trump at recent party gatherings and do not want to be associated with his sometimes-angry tone, according to advisers and close associates who have spoken with these Republicans.
Several Republican consultants said their clients were concerned that Mr. Trump’s unusually high unfavorable ratings with all voters and his unpopularity among women and Hispanics could doom him as a general election candidate and damage their own future political prospects if they were on his ticket.
Still, elected officials do have a way of coming around to the vice presidency, and Mr. Trump said in an interview on Saturday that he was in the early stages of mending fences and building deeper relationships with leading Republicans. And in a sign of growing acceptance that Mr. Trump is their likely nominee, several Republicans made it clear that they would join him on the ticket because they think he can win, or because they regard the call to serve as their duty.
Newt Gingrich, the former House speaker, as well as Senator Jeff Sessions of Alabama and the retired neurosurgeon Ben Carson, said in interviews that they would consider joining the ticket if Mr. Trump offered. Two governors, Chris Christie of New Jersey and Mary Fallin of Oklahoma, have also told allies that they were open to being Mr. Trump’s running mate.
“If a potential president says I need you, it would be very hard for a patriotic citizen to say no,” Mr. Gingrich said. “People can criticize a nominee, but ultimately there are very few examples of people turning down the vice presidency.”
Mr. Trump, who could well become the presumptive Republican nominee on Tuesday by winning the Indiana primary, is just starting to mull vice-presidential prospects and has no favorite in mind, he said in the interview. Mr. Trump said he wanted someone with “a strong political background, who was well respected on the Hill, who can help me with legislation, and who could be a great president.”
He declined to discuss potential picks in any detail, but he briefly praised three governors as possible contenders — Mr. Kasich, Mr. Christie and Rick Scott of Florida — and said he would also consider candidates who were women, black or Hispanic. (A spokeswoman for Mr. Scott said he was focused on being governor.)
Asked if he was surprised about the array of Republicans who are uncomfortable being his running mate, Mr. Trump said: “I don’t care. Whether people support or endorse me or not, it makes zero influence on the voters. Historically, people don’t vote based on who is vice president. I want someone who can help me govern.”
A cross section of leading Republicans agree that his most sensible choice would be an experienced female governor or senator, given that he would most likely face Hillary Clinton in November and need support from a majority of white women to offset her strong support among blacks and Hispanics. Yet Mrs. Clinton is currently ahead of Mr. Trump with white women by double-digit percentages, according to a recent CBS poll.
The pool of Republican women in major offices is relatively small, and Mr. Trump has already alienated some of them. Governor Haley denounced him for not quickly disavowing support from the former Ku Klux Klan leader David Duke, and Governor Martinez has criticized his remarks about Hispanics.
Both governors endorsed Senator Rubio for president; a Martinez spokesman said she “isn’t interested in serving as vice president,” while a Haley spokesman declined to comment.
“There are some Republicans who would’ve said yes to running with Romney or McCain or Bush but would say no to Trump,” said Curt Anderson, a Republican strategist, referring to the party’s last three presidential nominees. “The issue is, no one knows what we’re dealing with here. Is it possible that Trump faces a historic landslide loss? Sure. Is it possible he beats the hell out of Clinton? Sure. No one knows — no one has predicted Trump right for a long time.”
Even Governor Fallin of Oklahoma, who has not ruled out running with Mr. Trump, has expressed uncertainty about what he would be like as a leader, according to close associates who have spoken to her. Ms. Fallin, in a brief statement, would not discuss Mr. Trump, but said the nation’s challenges were too great for “business as usual” political solutions. “Any discussion of other service I might be asked to offer to my country is flattering but premature,” she said.
David Winston, a veteran Republican pollster, said Mr. Trump’s first challenge in finding a running mate was lowering his unfavorability ratings of 60 percent or more, because prominent politicians would not want to join his ticket if he cannot turn those figures around. Mr. Winston dismissed the notion — put forward by some Trump advisers — that the candidate could improve his ratings by picking a woman, a Hispanic, or other figure with demographic appeal.
“He simply won’t be able to convince any top-tier candidate to run with him if he can’t get those unfavorable numbers down,” Mr. Winston said.
Mr. Trump’s best hope may be Republican enmity for Mrs. Clinton, some Republicans strategists said. They predicted that Mr. Trump would ultimately have more options than his skeptics might assume because Republicans will ultimately unify in June and July with a deep and shared determination to beat her, and the traditional thrill of being considered for vice president could then kick in.
“I think he may have more choices than many people would suspect, because a lot of people will be flattered to be asked,” said Russ Schriefer, a Republican adviser to the Romney campaign in 2012 and to Mr. Christie during his 2016 presidential bid.
Mr. Schriefer emphasized that he had not talked to Mr. Christie about the vice presidency, but other Christie confidants said that he supported Mr. Trump strongly and would be willing to consider the No. 2 spot. A Christie spokesman, asked about the governor’s willingness, pointed to Mr. Christie’s response about the vice presidency at a recent news conference, where he said he would evaluate the offer “for any position in government.”
As a political novice, Mr. Trump will be widely judged on whom he chooses — and how and why he chooses the person — because voters and other Republican leaders will look to his pick to evaluate his priorities for the kind of advisers he would want as president.
“This is a big deal because it’s the first major decision he’ll be making as the nominee, and it’s important that the American public see his decision-making process and how he goes through making such a big decision,” said Scott W. Reed, the U.S. Chamber of Commerce’s senior strategist.
Other than elected officials, Mr. Trump also said he was open to people with deep national security experience — which some Republicans think should be his top criterion.
“What Donald Trump needs is the most experienced, most qualified foreign policy mind in Washington, and somebody that would immediately bring calm to the choppy political waters that always seem to be around him,” said Joe Scarborough, a former Republican congressman from Florida who now hosts “Morning Joe” on MSNBC. He suggested Robert M. Gates, the former defense secretary, but was more circumspect when asked if he was willing to be Mr. Trump’s running mate himself.
“I definitely have a lot of strong opinions about who it should be. (Not me!!),” wrote Mr. Scarborough, who served on the House Armed Services Committee and who has a good relationship with Mr. Trump.
Other Republicans were more open about joining Mr. Trump on the ticket. Senator Sessions, who is advising Mr. Trump on foreign policy, said he would send his personal tax information to the Trump campaign if it wanted to vet him. Mr. Carson, who was a Republican presidential candidate and battled with Mr. Trump before dropping out and endorsing him, said he would prefer to remain an outside adviser to Mr. Trump, but added that he was willing to join the ticket if he would “bring something that other people wouldn’t bring.”