President-elect Donald Trump has narrowed his short list for secretary of State to four, and could pick his top diplomat this week, aides said Wednesday.

Many experts consider the post the most important of any incoming president’s Cabinet selections. Trump aides are quick to tell reporters the incoming commander in chief is eager to select the best person for the job rather than doing so quickly.

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The Trump aides said it would be the last Cabinet-level pick until at least next Monday.

Still in the running is former Massachusetts Gov. Mitt Romney, the Republican presidential nominee in 2012. He dined on frog legs, scallops and gourmet chocolate cake with Trump at an upscale Manhattan restaurant Tuesday evening. Aides said they discussed “the state of the world” and got to know each other better in their second face-to-face meeting in 10 days.

Two candidates who might encounter difficulties garnering enough votes in the Senate are former New York Mayor Rudolph Giuliani and retired Army Gen. David Petraeus, the former CIA director. Questions linger about Giuliani’s global business ties and Petraeus’ legal issues stemming from his sentencing in 2015 to two years probation and a $100,000 fine for providing classified data to his biographer and mistress.

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What’s more, Senate Republicans worry Trump and the GOP would look hypocritical for nominating and supporting Petraeus after candidate Trump blistered his Democratic general election foe, Hillary Clinton, for her misuse of a personal email server and classified information while she was the country’s chief diplomat.

That leaves Senate Foreign Relations Chairman Bob Corker of Tennessee as what some in Washington call the safe pick. He seemed to fall out of Trump’s orbit, before re-emerging this week in the secretary of State race.

But, notably, Democratic aides and sources have reached out to Roll Call in recent weeks to raise questions about Corker’s own criticized business dealings and disclosure missteps.

Contact Bennett at johnbennett@cqrollcall.com. Follow him on Twitter @BennettJohnT.