Sport, which has close ties to Barca, says president Sandro Rosell has upped the stakes in negotiations over the sale of Thiago Alcantara to United.
Rosell has suggested United also take David Villa as part of a package which would see Rooney move in the opposite direction.
The England ace would arrive as a direct replacement for Villa, so solving a dilemma for coach Tito Vilanova, concerned over losing Villa's goalscoring power.
The total deal would be Thiago, Villa plus €4 million cash to United for Rooney, who would be valued at €30 million.
Rosell has made no secret of his admiration for the England ace, who he has likened to Nou Camp legend Hristo Stoichkov.
The stumbling block to the deal is United manager David Moyes, who is yet to accept Villa as part of the deal.