The England international was booed towards the end of last season after reports emerged that he had asked to leave the club following a falling-out with then-manager Sir Alex Ferguson.
Robson believes that the situation is similar to the one in which former United player Cristiano Ronaldo found himself back in 2006, when Rooney was sent off during England's World Cup quarter-final against Portugal which led to Red Devils fans fuming with the winger for winking at his teammate.
"If he starts the season, scores a few goals and enjoys his football, the fans will forgive him, if there's anything to forgive him for," Robson told reporters.
"After two games, Ronaldo was playing fantastically when he came back to Old Trafford. Everybody forgot about it and everybody loved Ronaldo.
"And I think that's the scenario for Wayne."
Both Arsenal and Chelsea have been linked with trying to sign Rooney.