The 18-year-old has one year left on his current deal – and the new contract is thought to be worth £20,000 a week
Rashford scored twice on his debut in February’s Europa League win over FC Midtjylland, then again in his first Premier League game three days later.
He is England’s youngest debut scorer and will find out on Tuesday if he is in Roy Hodgson’s Euro 2016 squad.
Rashford scored just 138 seconds into his first senior international appearance as England continued their preparations for next month’s tournament in France with a
2-1 win over Australia on Friday.
Since bursting onto the scene three months ago, Rashford has netted eight times in 18 appearances for the Old Trafford club.
The teenager has been represented in negotiations by family members, with advice from Chris and Wayne Welbeck, brothers of former United forward Danny, who now plays for Arsenal.
Meanwhile, it is thought United are also close to a new deal for defender Cameron Borthwick-Jackson, who has one year to run on his existing contract.
The 19-year-old made 14 appearances under Louis van Gaal this season.
Van Gaal was sacked two days after the club’s FA Cup final win over Crystal Palace and has been replaced by former Chelsea boss Jose Mourinho.