No and realistically there's nothing he can do to really put himself into conversation (otherwise he would've already been there, he's done enough).
As others have said, his competition, depending on the formation, would be mostly either attacking midfielders or second strikers, so he's competing directly against Sir Bobby (the greatest player in the history of this club), Cantona (arguably the most influential player of the entire Fergie era) and Rooney (arguably the best, especially if we include longevity, player of the Fergie era). He's really not a central midfielder to play in a midfield 2 if we go with this formation and there he'd be at least behind Keane, Scholes & Robson (probably Edwards as well — and then there are a fair few of other central midfielders that suit the role better despite not matching Bruno's overall impact on United's history).
Law
Best - Charlton - Cristiano
Scholes - Keane
Irwin - Vidić - Rio - Neville
Schmeichel
Rooney - Cantona
Giggs - Robson - Edwards - Beckham
Evra - McGrath - Stam - ?Brown?
van der Sar
He probably makes the third team though. And he'd be one of the best if not the best player on that team.
Hughes*
Whiteside - Bruno - Coppell
Carrick - Crerand
Byrne - Foulkes - Buchan - ?Rafael?
De Gea
* or van Nistelrooy/van Persie/Cole/Ole/Taylor... such an embarrassment of reaches.