As he keeps fighting the 2020 election results and refuses to concede to President-elect Joe Biden, President Trump remains quiet about the possibility of running to reclaim the White House in 2024.

“I don’t want to talk to 2024 yet,” the president told reporters on Thanksgiving.

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But behind closed doors, Trump has told advisors that he wants to run again in four years, and could potentially announce his bid before or even during Biden's Inauguration on Jan. 20, according to Fox News’ chief White House correspondent John Roberts, who confirmed a report from The Daily Beast.

Such an announcement, or even Trump’s flirtation with a third White House run, would definitely put a damper on what was thought to be a wide open battle for the GOP 2024 presidential nomination. And it could potentially freeze out early moves by other Republicans with national aspirations.

President Donald Trump dances as he walks off stage after speaking during a campaign rally at Miami-Opa-locka Executive Airport, Monday, Nov. 2, 2020, in Opa-locka, Fla. (AP Photo/Evan Vucci)

Republican strategist and GOP presidential campaign veteran Alex Conant says a Trump 2024 announcement would “absolutely” change the dynamic of the next White House race, “at least at the outset.”

“Nobody wants to be the first candidate to challenge Trump in 2024. If it were truly an open race, you’d see a lot of potential candidates making some aggressive early moves, going to Iowa, New Hampshire,” noted Conant, a founding partner of consulting firm Firehouse Strategies. “You would normally see candidates going through the early states before the end of this year.”

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While some potential 2024 GOP White House hopefuls – such as Sen. Tom Cotton of Arkansas, former ambassador to the United Nations and former South Carolina Gov. Nikki Haley, and South Dakota Gov. Kristi Noem - campaigned in Iowa and New Hampshire this autumn on behalf of the president and down ballot Republicans, nothing’s on the schedule so far for the rest of this year.

“2024 candidates don’t want to talk about Donald Trump. They want to talk about themselves. But so long as he is making noises about running again, the race will be all about him. And these candidates will be less inclined to do the early state travel because it will all be in the shadow of whether or not Trump runs again,” Conant highlighted.

Longtime GOP consultant David Carney agrees that with the president mulling a bid to return to the White House, it’s a far different scenario than the wide open fields for the GOP presidential nominations in 2008, 2012, and 2016.

But Carney, a veteran of Republican presidential campaigns for three decades, emphasized “I don’t think any serious candidate would be scared off because it’s four years. No one knows what’s really going to happen with the president. I don’t see it having much affect right now.”

Carney, the founder of the Norway Hill Associates communications and public relations firm, suggested that “maybe two dozen people are thinking of dipping their toes into the waters and see what happens.”

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Dan Eberhart, a prominent donor and bundler for the president and other Republicans, pointed to a waning in the president’s clout.

“Whether he likes it or not, the Trump ice cube is melting. He can announce a 2024 bid but the enthusiasm with donors is likely to fund the coffers of people like Cotton, Haley, (Sen. Josh) Hawley (of Missouri), or (Florida Gov. Ron) DeSantis to do battle with Trump,” Eberhart, the CEO of the oil drilling company Canary LLC, told Fox News.

Trump has made it clear that he intends to stay very involved in Republican Party politics after he leaves the White House in late January. He set up a leadership PAC immediately after this month’s election, and much of the fundraising the past few weeks for his legal moves to contest the election results are being steered to the new organization. Trump also endorsed Republican National Committee chair Ronna McDaniel – a major ally – for another term steering the national party committee.

The president is also heading to Georgia on Saturday, to campaign on behalf of the two Republican incumbents in the state's twin Jan. 5 Senate runoff elections, where the GOP's majority in the chamber is at stake.

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Trump took aim earlier this month at Gov. Mike DeWine, a day after the Ohio Republican called Biden the president-elect and said that “we need to begin” the transition between the Trump administration and the incoming Biden administration. And on Monday he slammed GOP Gov. Brian Kemp of Georgia as “hapless” for allowing a “sham” recount in the state’s presidential results, which upheld Biden’s narrow victory over Trump.

Both governors are up for re-election in 2022 and could be targeted by Trump.

Many Republican member of Congress have yet to urge the president to concede to Biden and have refused to recognize Biden as the election winner, out of apparent fear of the president’s strong backing from Republican voters.

Nearly two-thirds of Republicans in New Hampshire, the state that holds the first primary in the race for the White House, say they would like to see Trump make a 2024 bid to return to the White House. That’s according to a survey from the University of New Hampshire (UNH) conducted Nov. 19-23.

“Until Republicans come up with an obvious person to replace Trump, he’s going to be seen as the front-runner going into 2024,” noted UNH Survey Center director Andrew Smith.

And that’s a problem for other potential GOP presidential contenders.

“The last thing they want to do is have a big press conference in New Hampshire where they leave the impression that they’re going to challenge Donald Trump. Because what’s he going to do – he’ll start tweeting at them. It’s such a distraction for what they would normally be trying to do right now,” Conant noted.

In the absence of very public early state visits, Conant says White House hopefuls “will continue to do all the behind the scenes work – talking to donors, calling activists, they’ll do media, they’ll start to build out a team, but just have to be more deferential about it now.”

Fox News' chief White House correspondent John Roberts contributed to this report