Mr. Trump is deeply fond of Ms. Conway, whose job as counselor will give her frequent access to the president. Considered a “Trump whisperer” during the campaign, and someone who was particularly adept at explaining his appeal to voters, Ms. Conway will serve as one of the chief protectors of Mr. Trump’s political brand. She also has been a favorite strategist for other conservative candidates, such as Newt Gingrich during the 2012 presidential race, and she was an adviser to Mr. Pence during his run for governor of Indiana.

She has reportedly clashed at times with Mr. Priebus and Mr. Kushner. Both men were frustrated when she gave a series of interviews in which she criticized Mitt Romney, the 2012 Republican nominee, who was being considered for secretary of state despite having been a key figure of opposition to Mr. Trump during the campaign.

With a soft-spoken approach that has appealed to Mr. Trump, Ms. Conway was credited during the campaign — even by Democrats who opposed her — with smoothing out some of Mr. Trump’s most jagged edges during her appearances on television.

In the coming days, Mr. Trump is also expected to name three deputy chiefs of staff to help Mr. Priebus manage the sprawling operations at the White House. Among those being considered for these posts are Katie Walsh, who was the chief of staff at the Republican National Committee; Joe Hagin, who was the deputy chief of staff for operations through most of George W. Bush’s administration; and Rick A. Dearborn, who was the chief of staff to Senator Jeff Sessions of Alabama, who has been selected by Mr. Trump to serve as attorney general.

Tending to Mr. Trump’s relationships in Washington is likely to fall, in part, to Bill Stepien, who is in line to be the new political director. Mr. Stepien once played that role as the longtime political adviser to Gov. Chris Christie of New Jersey, but was fired for his part in the scandal involving a plot to punish a Democratic mayor by closing traffic lanes leading to the George Washington Bridge.

Mr. Stepien has been credited by Mr. Kushner and Mr. Bannon, who brought him on board in August, with helping them secure Mr. Trump’s victory in November.

It is unclear how Mr. Trump intends to use David Bossie, who left the conservative nonprofit group Citizens United to be the deputy campaign manager, and was among Mr. Trump’s earliest informal advisers about a presidential run. Mr. Bossie frequently clashed with Mr. Priebus, and Mr. Trump was dissatisfied with him by the end of the campaign. He has sought to be a deputy chief of staff.