(CNN) —  

A senior Republican senator warned the White House on Tuesday that it needs to do a better job consulting with the Senate before announcing controversial nominees like Herman Cain for the Federal Reserve Board, because it’s not “a given that everybody whose name gets sort of floated without vetting and without consultation can be confirmed.”

Sen. John Cornyn of Texas, a member of the GOP leadership team, predicted that Cain, a former businessman and Republican presidential candidate who faced sexual harassment allegations, will ultimately be cleared by the Republican-controlled Senate but said the “bigger issue” is that the White House can avoid problems by reaching out in advance.

“Rather than have the embarrassment for the nominees or for the President or for senators, the thing that I just encourage is there should be consultation,” Cornyn said.

President Donald Trump has said he wants Cain and Stephen Moore, a conservative economic and political commentator, who share Trump’s views on monetary policy, to fill seats on the Fed, which the President has routinely criticized. Democrats say neither man is a serious choice for the central bank, which is supposed to be independent of the White House.

Cornyn’s comments come amid a spate of openings for top jobs in the Trump administration, as the President has made many changes at the Department of Homeland Security and other agencies.

It’s not clear if Cain or Moore can be confirmed in the Senate, where Republicans have a narrow 53-47 majority. Senate Majority Leader Mitch McConnell notably did not make any effort to defend them when he was asked at a news conference if they were acceptable nominees.

“Well, we are going to look at whoever the President sends up. It’s his choice to decide and once he makes a nomination, we’ll take a look at it,” the Kentucky Republican said.

Sen. Richard Shelby, an Alabama Republican who’s a senior member of the Banking Committee, which would hold confirmation hearings for the pair, repeatedly noted to reporters that neither Cain or Moore had been officially nominated yet.

Shelby said Moore has “been around” and “active in politics.” But he seemed less comfortable with Cain.

“Cain’s nomination, if it comes around, will be interesting to watch,” he said.