Koeman had a falling out with United boss Louis van Gaal when the pair worked together at Ajax almost a decade ago, the Southampton manager feeling like his then director of football was undermining him.
"It is no issue. If we see each other, we shake hands. And that's enough," Koeman said when asked about their relationship. "The game is Southampton v Manchester United, and the game needs that attention.
"It is not about Ronald Koeman and Louis van Gaal. That is private. That was a working problem…it is not an issue, it is eight, nine years ago and now I prefer to talk about the game."
Van Gaal was similarly dismissive when approached with the same line of questioning as Koeman in his own press conference.
"I don't have to describe my relationship with the trainer of the opponent. I think that's private. We play against Southampton and we have to speak about Southampton and not their trainer," van Gaal said.
"Ronald Koeman is a very good coach and I know that he can do that. I think it's surprising that a club like Southampton is in that position now. That's always down to the trainer, the players and the staff. When you do that, it's very good."
The match is a vital one for both sides as they go into the weekend's fixtures separated by just one point, the Saints holding on to fourth place despite defeats by Manchester City and Arsenal in the last seven days.
Koeman remains positive in his side's ability but knows a win would bring added value after their recent blip.
"A win? It is important for the team," he said on Friday. "Always when you don't win some games, you need a win.
"If we play like we did last Wednesday we have chances to win on Monday. If we play like the last 20 minutes against City then we don't win on Monday.
"I believe in the way of playing and that is the most important thing because that has brought us all the points we have this season."