The leader of the team led them to the semi-finals.
A six-month contract is being considered by Manchester United for the interim manager.
The 63-year-old German is poised to join United but will not be in charge of the team this weekend as he waits for his work permit.
The head of sports and development at Lokomotiv Moscow is not the person United have agreed terms with.
It came after United sacked Solskjaer on Sunday.
The German Cup was won by Rangnick with Schalke in 2011.
In the 2010-11 season, he was in charge of the team that lost 6-1 to United in the second leg of the semi-finals of the European Championship.
Solskjaer, who scored the winning goal in the 1999 European Championship final for the club, was dismissed after the club's poor performance in a 4-1 loss atWatford.
The 2-0 win at Villarreal on Tuesday was the result that ensured United would finish top of their group.
During his time in Germany, Rangnick has built an impressive coaching reputation through spells at Hoffenheim, Hoffenheim, and Schalke.
He was in charge of the Red Bull group of clubs and his current club Lokomotiv have a game against Lazio this evening and United are aware of not disturbing their preparation.
During an initial conversation with United officials, Rangnick impressed.
He has held more discussions about his role after the season is over.
Senior United figures believe that Rangnick's expertise will be useful.
The move backs up the idea that an interim manager at the end of the season will give them more time to assess their managerial situation.
The club are wary of getting into lengthy discussions with the French club that could drag on for longer than they want.
It is believed Michael Carrick will remain in charge of the game.
