New York (CNN) President-elect Donald Trump is once again being pulled in two different directions.

In the same way he grappled with his choice for vice president, Trump is fighting competing impulses as he decides who to tap to be his White House chief of staff. The decision is a critical one that will shape Trump's presidency, the priorities his administration sets and how it deals with leaders in both political parties.

Trump's choice is coming down to two men who have closely advised the brash billionaire in the final months of his campaign, but whose backgrounds and political styles could not be more different: Republican National Committee Chairman Reince Priebus or Steve Bannon, the Trump campaign CEO and executive chairman of Breitbart News.

Priebus has the clear advantage.

Just as many of his advisers and party leaders urged Trump to tap Indiana Gov. Mike Pence of Indiana, the reliably conservative and mild-mannered counterbalance to the brash billionaire, top Republican officials are pressing Trump to tap Priebus for the top White House post.

House Speaker Paul Ryan and Senate Majority Leader Mitch McConnell urged Trump on Thursday in his first meetings in Washington as president-elect to pick Priebus, telling Trump they believed the RNC chairman is the right choice and the best fit for the position, a source told CNN on Friday.

Trump's son-in-law and top adviser, Jared Kushner, has also said privately he is supportive of that decision.

Priebus would be a reassuring presence to establishment Republicans still uncertain about what a Trump White House will look like. He would also bring decades of political experience and understanding of the wheels of power in Washington.

And while Trump ultimately asked Pence -- the favorite of conservative and establishment Republicans -- to join him on the GOP ticket instead of the bare-knuckled New Jersey Gov. Chris Christie, the newly-minted President-elect is once again being drawn toward making a pick who will enable his outsider persona and bellicose instincts.

Bannon is the very embodiment of those two characteristics, and he will also steer Trump toward enacting some of his most controversial and far-right policy proposals. And a source with knowledge of the transition told CNN on Thursday that advisers close to Trump believe Bannon is not suited to the top White House post, which demands keeping the wheels spinning and priorities aligned in the complex organization that is a presidential administration.

As the head of Breitbart News, Bannon has helped shape the alt-right movement for which the site heralds itself as a home and has helped drive hardline positions on immigration, terrorism and cultural issues to the center of the Republican Party. It's the same movement that Trump seized on early in his primary campaign that catapulted his improbable candidacy to the Republican National Convention.

The choice Trump now faces is one between two competing impulses -- both within his own mind and in the circle of advisers he listens to -- to fill the ranks of his White House with the hard-right figures who propelled his campaign to victory or the more establishment figures he will need to enact his policies in Washington.

Both Priebus and Bannon have traveled with Trump in the final months of his campaign.

Bannon has also been a major force behind some of Trump's more controversial stunts, including when Trump held an impromptu press event with women who had accused former President Bill Clinton of sexual assault and misconduct. He was spotted in the back of the room smiling as reporters were led in for the debate night surprise.

Yet Bannon would likely have a place in the White House regardless.

While at times Bannon is seen as feeding into Trump's worst instincts, he also has the credibility with the President-elect to convince him not to do things that would hurt him.

Priebus, though, earned Trump's trust in the final months of the campaign, as he steered the RNC's resources behind Trump's candidacy despite Republican officials urging Priebus to abandon Trump and instead help down-ballot candidates.

As Trump's candidacy was engulfed in the controversy stemming from a 2005 tape in which he boasted about being able to grope and kiss women without their consent, and as sexual assault allegations surfaced the next week, Priebus stood fast behind Trump and worked to salvage his campaign.

And when Trump took the stage in the wee hours of Wednesday morning to declare victory in the presidential transition, he ceded the podium to one man only, to thank him for getting him to the victorious moment.

That man was Reince Priebus.