Manager Louis van Gaal said the injury would need further investigation after the Spaniard was forced to sit out the last-32 first-leg tie where United's troubled season took another turn for the worse.
Sergio Romero replaced De Gea in the team for the Old Trafford club who were already without 12 players through injury or illness including captain Wayne Rooney.
"David de Gea has a knee problem so we have to wait and see. We will investigate that with a scan," Van Gaal said.
"When you play with another team every week, with other players, it doesn't help."
With United struggling to make the top four of the Premier League, the Europa League presented them with their best chance of qualifying for the Champions League next season.
The visitors took the lead against Midtjylland in the 37th minute with a scrappy goal from Memphis Depay.
Their advantage was wiped out a minute before halftime when Pione Sisto pounced on a mistake from Michael Carrick and scored with a low shot.
United were undone again in the 77th minute by substitute Paul Onuachu who swivelled before firing home a low bouncing drive to earn the Danes victory.
It was a second straight defeat for United who lost to Sunderland by the same score in the Premier League on Saturday.
"A lot of things are happening in the heads of the players," Van Gaal said.
"The first goal is deflected, it is unbelievable, it is too much. In the second half we didn't win any duels, any second balls.
"Jesse Lingard had two remarkable chances," added Van Gaal of United's young forward. "In certain phases we were not very well organised."
The return leg is next Thursday at Old Trafford.
(Reporting by Neville Dalton and Toby Davis; Editing by Tony Jimenez)
