The Red Devils go to Leipzig just needing a point to progress to the knockout stages, but defeat would almost certainly see them drop out of the competition.
Solskjaer will have to do without both Anthony Martial and Edinson Cavani for the crucial contest, with both struggling with minor knocks, but bad enough injuries to stop them travelling.
Both forwards started the Premier League game against West Ham on Saturday, but Cavani was replaced at half-time and Martial was taken off in the second half.
However, two players that missed that clash with the Hammers will be available on Tuesday.
‘David is travelling, Luke is travelling. Edinson and Anthony only minor niggles, but won’t be making the trip,’ Solskjaer told a press conference on Monday.
De Gea will almost certainly return to the starting line-up to replace Dean Henderson, but Shaw is now battling it out with Alex Telles for the left-back position and is not guaranteed to come straight back in.
Solskjaer could even field both men, with Shaw playing on the left of three centre-backs and Telles operating in the wing-back role.
Man Utd 23-man squad vs RB Leipzig
De Gea, Henderson, Grant, Bailly, Fosu-Mensah, Lindelof, Maguire, Shaw, Telles, Tuanzebe, Wan-Bissaka, Williams, Fernandes, James, Lingard, Mata, Matic, McTominay, Pogba, Van de Beek, Ighalo, Greenwood, Rashford