Former Massachusetts governor and GOP presidential candidate Mitt Romney makes his way through supporters of Iowa State Senator and U.S. Senate candidate Joni Ernst on Saturday in Cedar Rapids, Iowa. (David Greedy/Getty Images)

Officially, Mitt Romney returned to Iowa, the quadrennial presidential proving ground, to give a boost to Joni Ernst. But at a closed-door breakfast fundraiser here Monday, the first question from a donor had nothing to do with Ernst’s Senate campaign.

“When you get elected to the Senate, your job should be to convince Mitt Romney to run for president again,” a donor told Ernst, according to several attendees. The Republican candidate said she would, while Romney laughed.

When Romney and Ernst gathered in a West Des Moines boardroom with about 40 agriculture executives Sunday night, one businessman after another pleaded with Romney to give the White House another shot.

And at a rally for Ernst in Cedar Rapids on Monday, the state legislator who introduced Romney said, “If his address was 1600 Pennsylvania Avenue, I would sleep a lot better.” After Romney and Ernst finished speaking, some activists chanted, “Run, Mitt, run!”

Romney, the 2012 GOP presidential nominee and now the tacit head of the Republican Party, visited Iowa as part of a feverish nationwide tour designed to help the GOP take control of the Senate. He has insisted that he is not interested in running for president a third time. But his friends said a flurry of behind-the-scenes activity is nudging him to more seriously consider it.

Former presidential candidate Mitt Romney is hitting the stump for Republicans this fall -- which reminded us of some of his most memorable moments during the 2012 campaign. Here's a look at some of Romney's worst gaffes and a few predictions he got right. (Julie Percha/The Washington Post)

Romney has huddled with prominent donors and reconnected with supporters in key states in recent months. Because of the vacuum of power within his party and the lack of a clear 2016 front-runner, confidants said Romney is grappling with this question: If drafted, would he answer the party’s call?

Further juicing the speculation was a Des Moines Register-Bloomberg News poll released over the weekend showing that Romney is the only potential 2016 candidate who would beat Hillary Rodham Clinton (D) among likely Iowa voters, 44 percent to 43 percent.

People in Romney’s vast political orbit who are waiting and wishing on him to launch another campaign said Romney has done little to quiet them and has been hazy about his plans following next month’s midterm elections.

Former Minnesota governor Tim Pawlenty (R), who briefly ran against Romney in 2012 before becoming a close ally, said he wants to see Romney give it another go.

“There is a feeling that the country missed out on an exceptional president,” Pawlenty said. “If he runs, I believe he could win the nomination and the general election. It’d be the right person at the right time, and I would encourage him to do it.”

Pawlenty noted that Ronald Reagan ran unsuccessfully for president twice before being elected in his third attempt “and was stronger for it.” In contrast with Romney, Pawlenty said, “the emerging class of Republican candidates is untested and unproven.”

Within Romney’s political network, there has been informal chatter about a third run since early 2013, according to people familiar with the discussions. It bubbled up in phone calls and at dinners and has gained steam this year. Requests continue to pour in for him to appear on the campaign trail, and advisers said he is eager to mount a multi-state fly-around swing before Nov. 4.

In Iowa, however, Romney seemed uncomfortable with the 2016 talk. At the West Des Moines rally, he spoke for only five minutes, criticizing President Obama on income inequality, foreign affairs and other issues. When reporters tried to question him afterward, he sneaked into a dark maze of cubicles.

He also said that now that he was no longer a candidate, he had a joke to share involving Obama, golfer Phil Mickelson and tennis great Andre Agassi.

As Romney told it, Obama shows up at a bank to cash a check without his ID. The teller asks him to prove who he is, saying that Mickelson proved his identity by hitting a golf ball into a cup and Agassi proved his by hitting a tennis ball at a target. “Is there anything you can do to prove who you are?” the teller asks.

“I don’t have a clue,” Obama replies in the joke.

The crowd ate it up.

Former aides and senior Republicans say Romney appreciates the GOP masses crowing that he was right about issues such as Russia and health care. But what really intrigues him, they said, are the vulnerabilities among top-tier candidates in the Republican field. If Romney moves toward a race, it would be because he sees a path to victory.

“It’s the market pulling him,” said Kent Lucken, a longtime friend and adviser who accompanied Romney to Iowa. “People look at Hillary as the likely Democratic nominee, and the party needs a strong leader who can stand up to her and who’s been through the process.”

Romney is returning to Boston on Tuesday for a dinner that he and his wife, Ann, are hosting for former campaign advisers and business associates. The event — to benefit neurological research at Brigham and Women’s Hospital — has Romney intimates abuzz.

Save-the-date notices have gone out for the third annual Romney policy retreat in Park City, Utah, in June 2015 — a signal that he wants a platform to promote his issues as the presidential primary campaign season gets underway.

Romney is also mingling privately with top donors who could fund a third campaign. Romney visited Sept. 23 with Joe Ricketts, a billionaire investor who finances the Ending Spending super PAC, at Ricketts’s palatial penthouse apartment covering the entire 78th floor of the Time Warner Center in New York.

On Oct. 6, Romney also took part in a GOP fundraising dinner at the Manhattan apartment of Woody Johnson, the New York Jets owner and former Romney campaign finance chairman. Several 2016 hopefuls gave presentations to the donors, while Romney served as a co-host and made no pitch.

At Johnson’s home, Romney and media magnate Rupert Murdoch spoke about Romney’s political future. According to two Romney allies familiar with the conversation, Romney was cagey with Murdoch but expressed concerns about the developing GOP field. Romney told Murdoch that he felt uneasy about the party’s non-interventionist drift on foreign policy and the base’s embrace of ideological hard-liners.

Many Romney boosters believe that his window of opportunity will be in mid- to late 2015, should Sens. Rand Paul (R-Ky.) or Ted Cruz (R-Tex.) ascend and party establishment types turn to Romney as a savior.

If former Florida governor Jeb Bush (R) opts out of a campaign, “there is going to be more pressure on Mitt to go,” said Tom Rath, an influential New Hampshire Republican.

At a luncheon this month in Atlanta to help GOP Senate nominee David Perdue, “people sat up and paid attention” to Romney, said Sen. Johnny Isakson (R-Ga.). “I pulled him aside afterward to thank him for coming. He said he’s not running, and I take him at his word. But I don’t think the door is entirely closed, and circumstances can change.”

That phrase — “circumstances can change” — has been repeated by many Romney backers since the former nominee used it to describe his own thinking about 2016 in a radio interview last month with Hugh Hewitt.

Spencer Zwick, Romney’s former national finance chairman, talks regularly with Romney and said he has been receiving daily calls from donors and other supporters. “There are still plenty of donors who hope circumstances will change and there will be an opportunity for Romney to run again,” he said.

Zwick is part of a slimmed-down inner circle, including longtime advisers Beth Myers, Peter Flaherty, Stuart Stevens, Lanhee Chen and aides Kelli Harrison and Matt Waldrip, who are advising Romney on political activities this fall.

Romney traveled through Iowa with three trusted advisers and friends: David Kochel, Ron Kaufman and Lucken. He also reunited with supporters from campaigns past. In Cedar Rapids, Romney spotted Jim Wilson, a Virginia man who logged more than 40,000 miles chasing the GOP nominee from coast to coast in his campaign-festooned GMC pickup. The two hugged. “You son of a gun,” Romney said.

Another fan, Gary Chidester, 64, came to the West Des Moines rally with a full coterie of Romney paraphernalia for the former candidate to autograph: campaign placards, enlarged photographs and buttons of Mitt and wife Ann, and paperback and audio copies of Romney’s book “No Apology.” He also held a framed drawing that a friend gave him of a black cruise ship named Obama sinking into the sea and a white ship named Romney with the caption, “We’re here to save you.”

“He’s the only qualified person to run this time,” Chidester said. “Mitt is a business genius. That’s why I’ve listened to this tape three times. He had it all down — he had Russia down, he had the debt down — and all the other Republicans are novices by comparison.”

Costa reported from Washington.