Rooney, 28, is considering an offer that would make him the highest paid player in United's history.
The England striker is in the final 18 months of his £250,000-a-week contract and Chelsea boss Jose Mourinho remains interested in signing him.
United and Rooney hope to conclude contract talks in the coming week.
Rooney, who joined United from Everton in August 2004, is the fourth-highest goalscorer in United history.
However, according to former United manager Sir Alex Ferguson, Rooney asked for a transfer towards the end of last season.
He was linked with a move to Chelsea last summer and the Blues had a bid rejected by United.
Rooney scored 11 goals in 24 games for United this season before being sidelined by an injury which has kept him out since 1 January.

It follows on from the hugely successful first movie, reuniting stars including Robert Downey Jr as Iron Man and Chris Hemsworth as Thor.
Marvel villain Ultron is a rebellious artificial intelligence which creates an android that attacks the Avengers.
Whedon revealed the name after footage of the latest Thor film was shown.
The footage was introduced by actor Tom Hiddleston - who played villain Loki in the first Avengers film - in full costume, according to Hollywood Reporter.
Hiddleston stayed in character, making his speech and rallying members of the audience as they chanted the villain's name.
Comic-Con, which has been running since 1970, celebrates "the historic and ongoing contribution of comics to art and culture".
It holds conventions and events featuring comics and related popular artforms - not least films featuring comic-book characters and computer games.
Manchester United have opened talks with Wayne Rooney over a new deal that could keep the striker at Old Trafford for the rest of his career.
[NEXT_CONCEPT]
Director Joss Whedon has revealed the title of his 2015 Avengers sequel - Avengers: Age of Ultron - at the Comic-Con convention in San Diego.