"The way the president has basically designated someone who is not Senate-confirmed to oversee the investigation, someone who's already expressed his reservations about the investigation, that's a problem," Flake told The Hill.

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Flake hedged Friday on if recusal calls were the right step, instead circling back to concerns that Trump had picked someone not confirmed by the Senate to become acting attorney general and get oversight of the Mueller probe.

He countered that a president "being able to hand-pick" an individual, who has not been confirmed by the Senate, to oversee the investigation "doesn't seem right."

"I don't know if recusal is the right thing, but to have somebody who is not Senate-confirmed in a position to oversee, you know, an investigation of the president's campaign, that just does not sit well here and I don't think it's consistent with practice and the Constitution really," Flake told The Hill.

The Justice Department confirmed this week — after Trump announced that Whitaker would succeed Sessions in an acting capacity — that he would also be given oversight of the special counsel investigation.

"The leader said there's no urgency because the special counsel is in no danger, people aren't being fired. That's changed, obviously," Flake said about the floor tactics.

In addition to giving Mueller a path for potential recourse if he is fired, it would also codify that only a senior Senate-confirmed Justice Department official could fire him.

Flake opened the door Friday, in an interview with The Hill and Politico, to trying to insert the bill into a must-pass, end-of-the-year government funding measure. It would be an uphill effort, but if all Democrats and Flake were able to convince another Republican senator to join their effort they would have leverage in the negotiations.

"Yeah, I do. I do," he said, about trying to get it the spending bill. "The first step is to have this vote, and to have an affirmative vote and then obviously the next step is to put it in the spending bill."