Rooney suffered a deep gash in his forehead during a collision with a team mate at the club's Carrington base on Saturday and needed stitches, the Sunday Mirror reported.
The injury could also see the 27-year-old striker miss England's upcoming crucial World Cup qualifiers against Moldova and Ukraine, the paper said.
England face Moldova at Wembley on September 6 before travelling to Kiev to play Ukraine on September 10.
Image: Wayne Rooney
Photograph: Alex Livesey/Getty Images
Super Negredo nods in to see nervous City past Hull
What you must not miss in the EPL this weekend
EPL's new managers: early signs don't enthuse
League Cup: Fellaini rescues Everton with extra-time winner
Milan face Barcelona yet again in Champions League