Rangnick is currently the technical director of Lokomotiv Moscow, having only joined the Russian Premier League club during the summer.
Sources say that advanced discussions between United and Rangnick are ongoing over a six-month contract which would then see the 63-year-old stay on in another capacity.
As reported by The Independent earlier this month, Rangnick is keen on remaining at Old Trafford beyond the end of this season.
United have been focused on appointing an interim manager to replace Solskjaer before undertaking the process of appointing his permanent successor later this season.
It is not expected that Rangnick will be in place by the time of Sunday’s Premier League trip to Chelsea due to work permit reasons.
Solskjaer’s time in charge came to an end last weekend following the humilating 4-1 defeat to Watford, with Michael Carrick stepping up to take caretaker charge.