Rooney was left out of Gareth Southgate's squad for Wednesday's 1-0 defeat to Germany and Sunday's World Cup qualifier against Lithuania, amid concerns over the lack of playing time he has earned under Jose Mourinho this season.
The 31-year-old striker has made only nine Premier League starts for United this season, scoring just five club goals, and his Old Trafford future is in doubt.
However, Scholes feels Rooney should remain part of the set-up at both club and international level, as he has valuable experience to pass on to both sides' younger players.
“Wayne has been a sensational footballer for England and Manchester United for nearly 15 years. He is greatly respected and I still think he has a part to play with United and England,” he said.
“He has great experience, he can pass knowledge on to young players and it's up to Gareth if and when he wants to use it.
“I don't want to talk about him going to another English club, I hope he gets himself back into the United team. He has a fight on his hands but the Wayne Rooney I know is someone who will face that challenge and try to do it.
“He still has a prominent role to play for England and United. There's been a lot of speculation over where he goes now but Wayne, with what he's done, has earned the right to do what he wants to do. I hope he does stay and fight.”
Southgate has denied that Rooney's international career is over, despite the striker's omission from his latest squad.
“He's absolutely [in my thinking],” he said. “I didn't quite understand why we were talking about tributes when he has still got every chance of being in this team.
“I'm conscious that we'll be talking about Wayne again and I'm hoping to be talking about the players who are here, but there's no reason why he is finished with us.”
Additional reporting by PA