But Southgate is adamant Rashford should stick with Mourinho, who he insists values the youngster enormously.
But the United academy graduate will have to make a tough decision if he fails to earn a place in Jose Mourinho's starting line up.
"If you don't mind losing a couple of minutes, I'm going to do something not for you [the media] but the United fans and I think I hold myself to the United fans and for them I am going to spend two minutes with you", Mourinho began.
"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", Southgate said.
"Some of the persons, everybody knows they have double salaries, they work also for the clubs and of course they are not independent and they conduct things in the direction they want, that's obvious, human, is natural, not ethical but I accept it", said Mourinho.
Mourinho failed to navigate the barricade successfully, though, and got his foot caught, leading to him falling to the ground.
"We really wanted to make Marcus feel important over this period", said Southgate, according to Sky Sports. Possibly Everton is the sort of club you are looking at, just below the top six, for someone like Rashford, similar to what Lukaku did.
Rashford made his United debut in February 2016 in the Europa League and scored twice.
United entered the worldwide break with a 2-0 win against Burnley but will face unbeaten Watford at Vicarage Road on Saturday in another hard test for the Red Devils.
It is easy to understand his frustration as he has looked one of Europe's brightest young talents since breaking onto the scene just under three years ago.
Like with so many others in the England squad, though, that confidence can only be maintained if Rashford plays for his club, no matter what he has achieved so far and the impressive comparisons he has drawn. "More of a concern is the young players who might be coming through to challenge that are not getting an opportunity to play".
After Rashford scored against Spain and Switzerland, Carragher added: "I don't see Rashford ever displacing Lukaku so long as he is there".