Despite training on Wednesday, the 30-year-old captain will miss the club's European opener, as will Luke Shaw, Antonio Valencia and Jesse Lingard.
Manager Jose Mourinho said Rooney had been rested in advance of Sunday's Premier League trip to Watford.
Phil Jones is also absent after suffering lateral knee ligament damage.
The injury could sideline the defender for up to a month.
Mourinho wants a 'fresh' Rooney
"Wayne has played every game since the start of the season," said Mourinho of his decision to leave his captain in Manchester for the Group A opener.
"He has also played 90 minutes for England and he played 90 minutes last weekend and I want him to be fresh for the next match. In the position he plays, I have lots of solutions."
The former Chelsea and Real Madrid manager admitted the Europa League was not a competition the club wanted to be in, but said it was one they would aim to win.
"It's not the big dream of every big player, but we are not in the Champions League," added the Portuguese.
"We have to look at it with respect. I think we want to do well in the competition and for a club of our dimension to do well is not to be out in the group phase."
Michael Carrick, Matteo Darmian, Timothy Fosu-Mensah and Memphis Depay could also feature for Mourinho's side after failing to make the squad for Saturday's 2-1 Premier League home defeat by Manchester City.
Feyenoord are taking various measures for European home games this season in a bid to avoid triggering suspended punishments handed down by governing body Uefa for crowd trouble.
For Thursday's match, the capacity at the De Kuip stadium will be reduced to 30,000 and additional stewards will be employed, while a 10m high net will surround the pitch.
Feyenoord have taken these measures in attempt to avoid Uefa sanctions following trouble at previous European games.
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