Rand Paul. Chip Somodevilla/Getty Images Sen. Rand Paul of Kentucky on Tuesday suggested he was not sold on the need for investigations into the mounting controversies roiling President Donald Trump's administration.

Paul said in an interview on the "Kilmeade and Friends" radio program on Tuesday that he didn't think it was prudent to launch "investigation after investigation" on fellow members of the Republican Party.

"I don't think it's useful to be doing investigation after investigation, particularly of your own party," he said.

He continued: "We'll never even get started with doing the things we need to do like repealing Obamacare if we're spending our whole time having Republicans — I think it makes no sense."

Several Republicans in Congress have called for a full inquiry into Michael Flynn, the former national security adviser to Trump. Statements Flynn made about conversations he had with a Russian ambassador before Trump's inauguration ultimately led to his resignation on Monday.

Senate Majority Leader Mitch McConnell on Tuesday said it was "highly likely" the Senate Intelligence Committee would dig deeper into Flynn. Other Republicans have echoed those sentiments, including Sen. Lindsey Graham of South Carolina. He asked on Tuesday: "Did Gen. Flynn do this by himself, or was he directed by somebody to do it?"

Sen. Bob Corker of Tennessee, who chairs the Foreign Relations Committee, said a full investigation "wouldn't be out of the question."

Democratic lawmakers overwhelmingly lambasted Flynn following his resignation and demanded answers about who else may be implicated in possible illicit communications with the Kremlin.