Rooney came off in the 88th minute of the heavy loss at Goodison Park with a knee problem.
United need their captain fit for the remaining four games of the season and manager Louis van Gaal is hoping the problem is not a serious one.
“Rooney was injured. He had a kick on his knee, so we have to wait and see how heavy it is,” Van Gaal said.
James McCarthy, John Stones and Kevin Mirallas found the net as United stumbled to their heaviest league defeat under the Dutchman.
A fortnight ago United seemed all but certain to qualify for the Champions League, but Liverpool are now seven points them and they have a match in hand.
“This is the first match that the other team have shown more fighting spirit than us” – Louis van Gaal
“You give the light to your opponents,” Van Gaal said when asked what effect the loss would have on United’s Champions League hopes.
Van Gaal has rarely criticised his players in the second half of the season, but it was a different story on Sunday as they were hammered despite having 65% possession and 17 goal attempts.
The United boss said he could tell his team were not up for the fight from their relaxed attitude in the warm-up.
“This is the first match that the other team have shown more fighting spirit than us,” Van Gaal said.
“Our motivation, inspiration and aggression was not as high as the opponent.
“I had already the feeling and my colleagues (assistants) Ryan Giggs and Albert Stuivenberg had the feeling because the warm up was not so good as usual.
“In the last minutes before the game you hope you can recover and stimulate your players but then it is too late.
“You have to prepare the match already two or three days before and Everton has done that and they have won because of that.
“When you see the goal after four minutes, three duels in a row we lost and that is an example. In the first half we created a lot of chances, five I think but we never scored but when you see the shots on the goal it is also without conviction.
“I hope it shall not happen again but as a coach I know it shall happen again.”
Van Gaal’s opposite number Roberto Martinez was much happier after the final whistle.
His team deserve praise for the way they contained United and then punished them so destructively on the counter.
And Martinez reserved special praise for Stones after he scored his first goal for the club.
“John deserves his moment because scoring his first goal is a memory he is going to treasure,” the Everton manager said.
“The boys were giving him a hard time because he only scores in training. His defensive display was his best since he was playing in the first team.”