Rashford has been playing through the pain barrier since suffering a muscle tear of his left shoulder against Arsenal last November.
The 23-year-old, who scored 21 goals and provided 15 assists in all competitions for the Red Devils last season, could go under the knife ahead of the 2021-22 campaign.
Rashford said: "It's either an operation or I don't get an operation.
"I'm not 100 percent sure if I need it yet, so I'll just take it as it comes. I'll finish strong here [in the European Championship] and I've got a few weeks off to just relax and decide what I'm going to do."
He added: "The doctors have not said yet. 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 am fully concentrated on here. When the tournament is finished, then I will ask the question."
Rashford has been limited to three substitute appearances in Euro 2020 so far and knows he has much more to offer ahead of a last-16 showdown with Germany at Wembley next Tuesday.
"Obviously I am not performing at my best and what I know I can perform to," Rashford said.
"Whatever that is down to, it doesn't really matter because it has been going on since early on in the season. I managed to get through the season with United.
"I think I had 36 goal involvements so I can't look back at the year and say 'mmm, I should have taken time off to do this and do that'.
"That's just not the way I look at things. I just want to be available for every game."