However, he has struggled to hold down a regular first-team place under Jose Mourinho at Old Trafford and will be suspended for United's next three domestic games after being sent off at Burnley before the global break.
The midfielder made his first start of the season against Switzerland having previously just come off the bench twice for Maurizio Sarri's men.
"They have a hard job to do and they have big squads at the top six, with huge competition for places".
The England manager hopes that the intensification of the domestic season, with some clubs competing on three fronts as Europe fully kicks in next week, will mean that they get opportunities.
A United source said: "Marcus loves the club and does not want to move but he wants to be a striker or at least play down the middle as a No 10".
"It's not my business to interfere with club managers, they have a hard job to do and they have big squads at the top six, huge competition for places, so Jose thinks the world of Marcus". He had left [Riyad] Mahrez out of the team, so he needed to get him on because he [Mahrez] is still adapting to the way of playing. "José [Mourinho] thinks the world of Marcus - he's a huge admirer of him - but he has his job to do". They have to try and win matches. They are going to have the League Cup and European football, so there's more opportunity that the squads will get rotated.
Meanwhile, Manchester United are preparing to return to Premier League action on Saturday evening when they travel to Watford.
Southgate, however, has reassured supporters that the young attacker, who scored the only goal of the game as England beat Switzerland 1-0 on Tuesday, is rated by his club boss as much as anyone else.
"His confidence will be boosted with two goals against top-quality opposition".
Romelu Lukaku has been United's undisputed first-choice striker since arriving in a £75 million deal from Everton in the summer of 2017, leaving Rashford to get the majority of his first-team minutes from the substitutes' bench or starting on the wing.