A growing number of Republicans are calling for Newt Gingrich to drop out of the presidential race to hasten the selection of a nominee and turn the country’s attention to the fall campaign against President Obama.

There is no sign that Gingrich is listening. The former House speaker flew to Illinois on Wednesday after defeats a day earlier in primaries here and in Mississippi, two Southern states where he had banked on big wins. He is scheduled to head to Louisiana this week, and possibly Texas. He has said he sees no reason to drop out, and his advisers say he is sincere when he promises that he will last all the way to the Republican National Convention in Tampa in late August.

Such doggedness worries conservative leaders who consider Rick Santorum’s tandem wins on Tuesday as more evidence that he is the preferred choice of voters who are seeking an alternative to Mitt Romney, the Republican front-runner. But Santorum needs Gingrich to step aside so he can face Romney one on one, several Republicans said.

“There won’t be a lot of public pronouncements, but there will be a lot of private phone calls beginning today,” said conservative strategist Keith Appell, who is supporting Santorum. “As much as Newt may feel that he should stay in, his campaign is really living on borrowed time.”

Richard Viguerie, another prominent Republican supporting Santorum, said Wednesday that he is “urging conservatives” to pressure Gingrich to step aside. He writes a newsletter that reaches 100,000 subscribers nationally, and he focused on the topic Wednesday.

“The vast, vast majority of people in the political arena, including almost all conservatives, don’t see a path for Gingrich to get the nomination,” Viguerie said in a telephone interview. “It’s hard to give it up, but I would expect after a few days he would give up. He’s never going to be stronger in the presidential race than right now. Every day that goes by, he’s going to lose support.”

Viguerie has no plans to make the case directly to Gingrich, and as of Wednesday, it was unclear whether anyone had. Several Republicans interviewed said there is a strong desire to give Gingrich the time he needs to make his own decision — and to afford him the respect due to a man who is considered a member of the family within the conservative movement. Santorum adviser John Brabender, for instance, said it is “premature” to talk about having conversations with the Gingrich camp — although he added that “we would be very interested in sitting down and talking, but that would be a decision for their campaign.”

One prominent conservative who spoke on the condition of anonymity to discuss the matter candidly said an effort is underway to ask Texas Gov. Rick Perry to urge Gingrich to step aside. But Perry spokesman Ray Sullivan said Wednesday that the governor is grateful to two of his top supporters — Louisiana Gov. Bobby Jindal and Kansas Gov. Sam Brownback — for sticking with him during his waning days in the race and for letting Perry decide when to withdraw.

Perry will give Gingrich the same courtesy, Sullivan said.

Romney supporters are more divided over whether Gingrich should quit. Some view him as helpful to Romney — because Gingrich helps split the conservative vote.

But others said they think Romney could outpace Santorum even without Gingrich as a foil. They are nervously watching the calendar as Republicans continue to battle one another rather than Obama. The time has come, they say, to rally around Romney.

Romney spokeswoman Andrea Saul said the decision to leave the race is up to each candidate — but she noted that a brokered convention would leave just two months “to take on an incumbent president who will have $1 billion to spend.”

“They’ll have to evaluate if that is really an option for winning the presidency,” she said.

Whether Gingrich’s departure would have any significant impact on Santorum or Romney is uncertain. Exit polling data suggest that Gingrich supporters would not necessarily flock to Santorum.

It also would be difficult for Santorum to win the nomination outright without a convention battle, even if all of Gingrich’s supporters did move to him. Santorum has 252 delegates to Romney’s 495, according to the Associated Press — and 1,144 are needed to win. Santorum would have to capture 66 percent of the 1,358 remaining delegates — a very high bar.

Gingrich’s more prominent supporters acknowledged that Tuesday’s results were not what they had hoped for but remained positive about their candidate’s decision to stay in the race. J.C. Watts, a former Oklahoma congressman and a Gingrich confidant, said he has no plans to urge Gingrich to call it quits.

“Gosh, he finished a strong second,” Watts said of Tuesday night. “Santorum hasn’t closed the deal with conservatives. Romney, the inevitable candidate, came in third.”

Bob Walker, another former House colleague, said a Gingrich departure would only hasten Romney’s nomination. “As long as both Gingrich and Santorum are in it, [Romney] is not going to get to 1,144,” Walker said. “If one or both of them drop out, he might get to 1,144.”