President Trump shakes the hand of Sen. Orrin G. Hatch (R-Utah) as he arrives for a speech at Utah State Capitol on Monday in Salt Lake City. (George Frey/Getty Images)

President Trump on Monday said he wanted Sen. Orrin G. Hatch to run for reelection, lavishing praise on the Utah Republican on several occasions during a brief visit to the snow-covered state.

"We hope you will continue to serve your state and your country in the Senate for a long time to come," Trump said in the State Capitol, calling Hatch a "fighter," a high compliment from the president.

At another event — while pushing a shopping cart through the Church of Jesus Christ of Latter-day Saints' Welfare Square as he purported to look for tuna — Trump mouthed "yes" when asked whether he wanted Hatch to run again in 2018. He seemed to throw cold water on a potential Mitt Romney bid, saying only that "he's a good man" when asked about the 2012 Republican presidential nominee. Romney was one of Trump's most vocal Republican critics during the campaign but later interviewed to be his secretary of state.

The future of Hatch, who is 83, has dominated political chatter in Utah and has sparked much behind-the-scenes maneuvering. Privately, Trump has told people he wants Hatch — chairman of the powerful Senate Finance Committee — to keep the seat and has implored the senator to run, saying he does not want Romney in the Senate. He also has told the senator that on several occasions.

Trump and Hatch have developed something of a rapport in the president's first year. While many Republican senators hold disdain the president — and some even criticize him publicly — Hatch speaks positively about Trump.

The president, in turn, has appreciated Hatch's public praise of his administration and children. The president also associates the Utah Republican with the success of the tax bill in the Senate.

On Monday, Hatch said Trump called him into the Oval Office on his fifth day in office and asked how he could help Utah. Hatch asked him to reverse the designation of 2 million acres of land in Utah as national monuments, the largest such reversal in history.

Trump was happy to comply.

"I asked for the president's help in fixing this disaster," Hatch said. "Without hesitation, he looked at me square in the eye and said we'll fix it."

He added that Trump was the "most famous man in the world." Trump seemed to like the introduction, saying it was "truly" a good one.

Meanwhile Utah Gov. Gary R. Herbert (R) didn't share Trump's fervent urging.

In an interview, Herbert said that Hatch "promised the people here five years ago that he wouldn't run again, and it would be his last term. For whatever reasons, he seems to be having second thoughts."

Herbert said that Hatch had been a terrific senator and a longtime friend, but "he's getting up there in age, and that causes concerns." He also said the polls indicate that while people love Hatch, they believe it "might be time to hang it up."

"The polls are a little difficult for him," Herbert said, adding it was Hatch's decision to make — and not his.

Herbert said Romney had been thinking about the Senate and "would be a tremendous asset to the Senate."

"He has a lot of talent and skills and more cachet than the freshmen senators," he said.

"I'm hearing in the ether they are both going to make the decision by the end of the year," Herbert said.