President-elect Donald Trump is considering a woman to run the Republican Party and an openly gay man to represent the U.S. at the United Nations, moves that would inject diversity into a Trump team.

The incoming president is considering Richard Grenell as U.S. ambassador to the United Nations. If picked and ultimately confirmed by the Senate, he would be the first openly gay person to fill a cabinet-level foreign policy post. Grenell previously served as U.S. spokesman at the UN under former president George W. Bush's administration.

At the same time, Trump is weighing whether to select Michigan Republican chair Ronna Romney McDaniel, a niece of Trump critic and 2012 presidential nominee Mitt Romney. She would be the first woman in decades to run the Republican National Committee.

"I'll be interested in whatever Mr. Trump wants," McDaniel told The Associated Press on Monday, adding that she was planning to seek the Michigan chairmanship again.

Appointing McDaniel to run the party's political arm could be an effort to help heal the anger after a campaign in which Trump demeaned women. The appointment of Grenell could begin to ease concerns by the gay community about vice-president-elect Mike Pence's positions on same-sex marriage during his time as Indiana governor.

Richard Grenell is being considered for the post of U.S. ambassador to the UN, which would make him the first openly gay person to fill a cabinet-level foreign policy post. (Osamu Honda/Associated Press)

The personnel moves under consideration were confirmed by people with direct knowledge of Trump's thinking who were not authorized to publicly disclose private discussions. They stressed that the decisions are not final.

Internal deliberations about staffing come a day after Trump made overtures to warring Republican circles by appointing RNC chair Reince Priebus as his White House chief of staff, and Breitbart News executive Stephen Bannon as chief strategist and senior counsellor.

The two men had made up the president-elect's chief of staff short list, and while Priebus received that job, Bannon is expected to wield significant clout. Trump gave top billing to the former media executive, who led a website that appealed to the so-called "alt-right" — a movement often associated with efforts on the far right to preserve "white identity," oppose multiculturalism and defend "Western values."

Priebus on Monday defended the media mogul, saying the two made an effective pair as they steered Trump past Democrat Hillary Clinton and toward the presidency. He sought to distance Bannon from the incendiary headlines on his website, saying they were written by unspecified others.

"Together, we've been able to manage a lot of the decision making in regard to the campaign," Priebus told NBC's Today. "It's worked very, very well."

Neither Priebus nor Bannon brings policy experience to the White House.

Chiefs of staff in particular play a significant role in policymaking, serving as a liaison to cabinet agencies and deciding what information makes it to the president's desk. They're often one of the last people in the room with the president as major decisions are made.

In announcing the appointments, Trump said Priebus and Bannon would work as "equal partners" — effectively creating two power centres in the West Wing. The arrangement is risky and could leave ambiguity over who makes final decisions.​