United are back in business after dismissing Jose Mourinho and placing Ole Gunnar Solskjaer in interim charge.
Under the Norwegian, the Red Devils have won nine of their 10 matches including games against Arsenal and Tottenham.
Solskjaer has made a strong case to be appointed as full-time manager after winning the hearts of the squad.
The club legend has brought out the best of players such as Rashford, Paul Pogba and Jesse Lingard.
Anthony Martial, whose future was in question, has already committed his future to the Old Trafford outfit after signing a five-year deal.
And United are keen to push forward with their plans to keep their best young players after opening talks with Rashford’s representatives, according to the Telegraph.
The 21-year-old has a contract that runs out at the end of next season, which has an option to be extended by 12 months to June 2021.
But United are keen to offer him a completely new contract that would see his wages double to £150,000-a-week.
It comes amid a month of interest from Real Madrid, who are on the peripheries should talks stall.
But the Telegraph go on to report that Rashford sees his future at the club he came through the academy of.
The Red Devils see the youngster as the poster-boy for their new era, even giving him the No.10 shirt to follow in the footsteps of past legends such as Wayne Rooney, Ruud van Nistelrooy and Teddy Sheringham.
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Rashford has been in incredible form under Solskjaer, scoring six goals including winners against Spurs and Leicester.
Against the Foxes, the England international netted with a fine finish early on to settle the game.
However, Solskjaer has revealed how Rashford ignored one instruction of his – but admitted that he has forgiven the Englishman.
The interim boss said: “I enjoyed Marcus’ finish.
“But then again I told him to try and go through Schmeichel’s legs because he opens up in these situations.
“But it was a great finish and the kid is improving and improving — so I can’t complain.”