During Tuesday afternoon's Senate Republican lunch, Majority Leader Mitch McConnell advised senators concerned about Trump's selection of former presidential candidate and pizza executive Herman Cain and conservative economic commentator Stephen Moore to share their views with the White House now, before Trump officially moves forward with the nominations, a source familiar with the remarks told CNN.

GOP senators are worried about various controversies swirling around the two men -- particularly Cain, who faced sexual harassment allegations that effectively ended his 2012 presidential campaign -- but also that Trump's nominees could weaken the independence of the Fed, the world's most powerful central bank.

Aiming to avoid a public fight over the confirmation, some Republicans hope to persuade Trump to reconsider ahead of Cain's official nomination.

It's unclear how effective the strategy will be at convincing the President, who has been outwardly enthusiastic about the picks, though top White House economic adviser Kevin Hassett earlier this week told reporters the nominations are still being vetted and are not done deals.

Trump said Wednesday morning that Cain "will make that determination" about whether to formally enter the confirmation process.

"Well, I like Herman Cain, and Herman will make that determination. Herman is a wonderful man, he's been a supporter of mine for a long time. He actually ran a very good campaign and that's up to Herman," the President told reporters outside the White House.

Trump continued, "Herman is, you know, he's already sat on one of the Fed Boards. And he's just somebody I like a lot, as to how he's doing in the process, that I don't know. You go through the process, but Herman's a great guy and I hope he does well."

Publicly, McConnell and other Republicans are projecting neutrality.

"Well, we are going to look at whoever the President sends up," McConnell said in a news conference Tuesday after the lunch. "It's his choice to decide, and once he makes a nomination, we'll take a look at it."

In hallway interviews with nearly 20 Republican senators on Tuesday, most were reluctant to candidly discuss Cain's potential nomination. Many said they hadn't heard the days-old news that Trump was planning to select him. Others opted to withhold comment, while some expressed vague reservations.

A few rejected the prospect outright.

"No," Sen. Cory Gardner of Colorado quickly responded when asked if he would support Cain.

"It's not about his past. It's about who I think should be on the board," the senator explained. "So that's that."

Republican Sen. Mitt Romney of Utah has been openly critical of the idea as well.

"I don't think Herman Cain will be on the board of the Federal Reserve," the freshman senator told reporters Tuesday. "It's important that the board be comprised of people who are academics, economists, and not people who are highly partisan."

Sen. Pat Roberts, like the majority of his colleagues, was slightly more diplomatic.

"When you first meet Herman Cain, he's very impressive," the Kansas Republican said. "But whether he's qualified to be on the Fed, I just don't know."

Republicans are wary of Cain, a former CEO of Godfather's Pizza who served as chair of the Kansas City Fed, for a variety of reasons. Four women have said Cain harassed them during the 1990s, when he led the National Restaurant Association. Cain denied the allegations, most recently in a video posted last weekend.

"It's all just sort of a vague memory right now, but it's hard not to hear 'Herman Cain' in the context of something like this and not immediately recall some of those challenges," Sen. Kevin Cramer said of the harassment allegations. The North Dakota Republican told reporters those allegations would factor into his decisions as a member of the Senate committee that evaluates Fed nominees.

Opponents also take issue with Cain's founding role in the pro-Trump political action group America Fighting Back.

But several GOP senators on Tuesday pushed back on critics, saying Cain was qualified for the job.

"He has an impressive background in business and understands the importance that our economy plays in everyday lives," Sen. Tim Scott of South Carolina told CNN.

Scott shrugged off the idea that Cain is an unconventional choice: "This has been an unconventional presidency, so that's not unusual."

Senate Finance Chairman Chuck Grassley of Iowa told reporters Tuesday that he wouldn't make up his mind until after a confirmation hearing, but he didn't have "any reason not to" support Cain.

Republicans are in a slightly better position to confirm contentious appointees than they were before the 2018 midterm elections, with their new majority of 53, but they can still afford to lose only a few votes before any nomination would be doomed.

That concern prompted Sen. John Cornyn of Texas on Tuesday to warn the White House to consult in advance with Senate leaders on nominations like these.

Trump's endorsement of Cain came just weeks after the President announced he would nominate Moore, best known as a Wall Street Journal commentator, to the seven-member board. (Moore was also a CNN contributor.)

The selection drew fire from Democrats, who said Moore was too partisan and was unqualified for the job. The two nominees' backgrounds in conservative politics have stoked fears about politicizing the Fed, even among Trump's GOP allies in Congress, after months of the President publicly railing against Jerome Powell, the Fed chairman he appointed, and the board's decision to raise interest rates late last year.

"I am concerned about that," Sen. John Kennedy, a Louisiana Republican, said on Tuesday. "The worst thing you can do in terms of your economy is to politicize or give the appearance that you're politicizing your central bank."

Moore has long run in Republican circles. He helped start the conservative Club for Growth and is a visiting fellow at the Heritage Foundation. He has recently been a vocal critic of the Fed leadership, siding with Trump on keeping interest rates low.

And he has already garnered support among a number of leading GOP senators, who say they believe he will win enough votes -- something they won't say about Cain.

"Well, I know him better and I expect he'll be confirmed," Sen. Roy Blunt, a Missouri Republican, said of Moore, after sidestepping questions about Cain.

Senate Appropriations Chairman Richard Shelby echoed that sentiment, praising Moore.

"Stephen Moore, I know well. I've worked with him. He does have a lot of experience in the area," said Shelby, an Alabama Republican.

As for Cain's chances, he added, "That will be a more interesting nomination."