ESPN revealed in April that the 23-year-old has been playing through pain for the last two years with shoulder and foot problems.
Rashford scored 21 goals and registered 11 assists for United last season but is yet to start any of England's matches at Euro 2020, making three substitute appearances totalling 58 minutes.
The forward is pushing to play some part in Tuesday's knockout clash against Germany at Wembley but has said he will determine whether to undergo surgery once England's involvement in the tournament is over.
"I have to decide after. It is either an operation or I don't get an operation," he said. "Probably [on] my shoulder but I'm not 100% sure if I need it yet. So, I will just take it as it comes. I will finish strong here and like I said I have got a few weeks off to just relax and decide what I am going to do."
An operation could potentially mean Rashford misses the start of the 2021-22 Premier League season, which begins for United with a home game against Leeds on Aug. 13 but Rashford added he is yet to discuss possible timescales with the club's medical team.
"The doctors have not said yet," he said. "The reason I don't know is that when the season was going, and before I came here, I knew there was no chance I was going to be getting the operation so I did not know how long the operation would put me out for.
"I never bothered to ask the question. I am fully concentrated on here. When the tournament is finished then I will ask the question."
England's upcoming game with Germany evokes memories of previous era-defining clashes, with the Germans knocking their old rivals out of tournaments in 1990, 1996 and 2010 but Rashford insisted this young group should not be burdened by that past.
"We are being put in a position where we are blessed to be part of that history," he added.
"Our main focus is to win the game, but if we do it gets put in history. That's how great teams are remembered for many, many years.
"There is no point fearing the past. You can't go back and change it. What we can change is the result of the next game and put ourselves in the best possible position to win the game.
"Some of the players weren't even born [in 1990 or 1996]... No, for us, Germany are a top, top team and have showed that for years but we can make play against them, we can give them a good game, and definitely over the last two years we've improved massively against the bigger teams, we've been getting better results.
"Whereas, before that, we were sometimes playing alright, but most of the game they were dominating, it was difficult for us to get goalscoring opportunities, and the game would usually be one or two-nil to them, but they'd be in control of the game.
"That was one of the things Gareth wanted to change. It was around Nations League time [in 2018 after the World Cup], we wanted to start pushing ourselves and trying to beat these big nations, and gradually we started to do it a few more times, sometimes drawing, but games where it's very fine margins, and losing a lot less.
"Anyone who is a fan of football, you get built up for these type of games and rightly so. In camp, it is different because as a player you have to be focused. You don't have time to get involved in that build-up because there's just too much emotion in the game and if you go on the pitch with all that build-up, we could end up with eight men on the pitch.
"So, yeah, we're just concentrated. We are going to have a proper training week and hopefully be at our best come the game."