Salt Lake City (CNN) President Donald Trump has taken an intense interest in whether Utah Republican Orrin Hatch runs for Senate re-election in 2018, sources tell CNN, a fact that has some in Washington thinking the concern is part of an effort to keep former Republican presidential nominee and on-again-off-again Trump critic Mitt Romney from the legislative body.

The renewed focus comes as Trump visits Utah for the first time as president. He made the trip as part of a Hatch-requested event at the Utah Capitol, where the President announced significant reductions to Bears Ears National Monument and Grand Staircase-Escalante National Monument. The President's decision is a nod to Hatch, who had made the issue a focus.

Trump praised Hatch during the event and said he hopes the senator will "continue to serve your state and your country in the Senate for a very long time to come."

Hatch offered Trump a colorful introduction, too, playing into the President's vanity by calling him "the commander in chief, the master deal maker and a wildly successful billionaire."

A White House official to Trump acknowledged on Monday that there is no love lost between Trump and Romney but said the idea that the President is pushing Hatch to run again because he wants to thwart the former Massachusetts governor is wrong.

"We like Sen. Hatch," the source said. "He was a critical voice in support of the tax reform debate. We like his voice in the Senate."

On whether Trump is in Utah as a subtle knock on Romney, the official said Romney was "not a focus" of Monday's trip.

"When the guy forms an exploratory committee, we will have a view," the official said.

Another source with knowledge of Trump's conversations with Hatch also downplayed the idea that the President's interest in the race had anything to do with Romney.

Trump has never spoken to Hatch about Romney, the source said, and the President began urging Hatch to run again long before Romney entered the conversation as a possible Senate candidate in October.

"President Trump has asked Hatch to run all year," the source added, noting that Trump's reasoning has primarily been the President's desire to score legislative victories on tax reform, health care and judge nominations.

In March, Hatch told CNN that Trump was urging him to run again.

Deeply torn

Hatch is deeply torn about running, several sources close to the 83-year-old senator tell CNN. He feels that he still has unfinished work to do in the Senate, and his ability to shepherd through tax reform -- which has long been a top goal for him -- has deepened his feeling that he can continue to be an effective and powerful leader on Capitol Hill at a time when his party is deeply divided.

Hatch has long been known for his eagerness to engage in bipartisan alliances, with the late Democratic Sen. Ted Kennedy of Massachusetts, for example. That kind of cooperation is a rarity in the polarized world of Washington these days, which is one more reason Hatch thinks that it could be important to stay in his seat.

As Hatch wrestles with his decision, Romney is being widely encouraged to run for the seat, should Hatch retire. Several "draft Romney" groups have formed across Utah trying to nudge the former Republican presidential nominee into the race. (None of them, however, have the blessing of or cooperation with Romney's nuclear team of advisers, according to several sources who speak regularly with the former Massachusetts governor).

In discussions with nearly a dozen friends, donors and former advisers, Romney has expressed strong interest in the post if Hatch were to retire. One source familiar with his thinking said Romney is "mentally there" in terms of being willing to run, but is waiting for Hatch to decide. Despite his interest, Romney allies are being extremely cautious and deferential to Hatch -- concerned that any inartful wording would suggest that they are trying to push him out of the race.

A leading Romney adviser would say only: "Sen. Hatch will announce his decision, which he has not done yet. Mitt Romney has neither opened the door nor closed it."

Prickly relationship

The prospect of Romney running for Senate, though, rekindles a focus on his prickly relationship with Trump.

Trump and Romney met twice shortly after the businessman-turned-politician won the White House in November 2016. The two had a warm and animated conversation at a table at Jean Georges at Trump's New York hotel in November and also met at Trump's golf course in Bedminster, New Jersey. Romney, at the time, was under consideration for secretary of state. Trump eventually chose Rex Tillerson instead.

Romney has been critical of Trump in the past. In March of 2016, he slammed Trump as a "phony" and a "fraud" in a speech in Utah.

"Here's what I know. Donald Trump is a phony, a fraud. His promises are as worthless as a degree from Trump University," Romney said. "He's playing the American public for suckers: He gets a free ride to the White House and all we get is a lousy hat."

His criticisms have continued into the Trump White House. After Trump blamed "both sides" for inciting deadly violence in Charlottesville, Virginia, earlier this year, Romney said Trump's remarks "caused racists to rejoice."

Trump confidants and White House aides think Romney would likely be critical of Trump in the Senate, adding his voice to a growing number of Republicans -- many of whom are not running for re-election in 2018 -- who are more forcefully speaking out against the President.

The tension between Trump and Romney has filtered down to their aides, too. According to the Washington Examiner , former White House strategist Steve Bannon is considering endorsing Hatch to keep Romney out of the Senate.

The speculation around Romney comes as there is confusion in Utah -- and among Romney confidants -- about what Hatch will decide.

As an example, several Romney backers pointed to the fact that Hatch recently added four staff members in Washington. He has also been raising money for his possible re-election campaign. The senator raised $3.4 million between January and the end of September, according to Federal Election Commission reports.

Romney would have few barriers if he wanted to get into the race. Hatch spent $10.6 million during his last re-election, in 2012, and Romney has the national name identification and fund-raising prowess that would make it difficult for another Republican to challenge him.

He already has a clear runway to the seat, too, should Hatch retire: Boyd Matheson, a prominent Utah Republican and former chief of staff to Sen. Mike Lee, said he would not run for the seat earlier this year. Given Romney's soaring popularity in Utah, several sources said, Matheson was not interested in engaging in a race that he was likely to lose.