The Premier League giants have identified the German, 63, who is currently head of sports and development at Russian club Lokomotiv Moscow, as a potential short-term successor to the Norwegian.
Michael Carrick took charge of the side for Tuesday's Champions League win at Villarreal, which secured progress to the knockout stages, but the club want to bring in a more experienced figure until the end of the season.
United are planning to make a permanent appointment at the end of the campaign, with Paris Saint-Germain boss Mauricio Pochettino strongly linked to the job.
The Athletic reported that Rangnick, the former RB Leipzig manager, had reached agreement with United to take over, saying the agreement was subject to Lokomotiv agreeing to release him from his contract.
But it said he would not be in charge in time for the game at Chelsea this weekend due to work permit issues.
Solskjaer was sacked on Sunday following his side's embarrassing 4-1 defeat to lowly Watford.
They are eighth in the Premier League after losing four of their past five matches, 12 points behind leaders Chelsea and six points adrift of fourth-placed West Ham.