The two sides are separated by just a point in the upper reaches of the Premier League, although Koeman's side have lost their last two games against Manchester City and Arsenal.
Two of Southampton's four league defeats have come in the past week and Koeman's countryman and opposite number at Old Trafford Louis van Gaal will see in-form United leapfrog Monday's opponents with a win.
Koeman - who is without Jack Cork (ankle), Morgan Schneiderlin (groin) and James Ward-Prowse (ankle) - warned that Van Gaal's men should not expect an easy ride, suggesting United may not be considered favourites as has been the case in previous seasons.
The former Feyenoord coach also dismissed suggestions there would be any ill-feeling between himself and Van Gaal despite disagreements during their time at Ajax.
"If we [Koeman and Van Gaal] see each other we shake hands. He's a great coach, it was difficult at the start but I think they are on the way back," he said in midweek.
"They have shown that in the performances of the team and have won the last four games, it's a nice challenge for us.
"Normally, if Southampton play Manchester United, Manchester United is the favourite, but they know it will be a tough game.
"Our performance against Arsenal was a good one, much better than the City game, but in the Premier League, there isn't an easy game to win."
While Southampton come into Monday's meeting following back-to-back defeats, United will target five Premier League wins in a row - aided by the return of captain Wayne Rooney.
The striker missed the midweek trip to Stoke with a knock to the knee suffered in the win over Hull City last weekend, but has been passed fit to return.
Van Gaal - who can also call on defender Jonny Evans again after he recovered from a foot injury - is not paying too much attention to Southampton's lofty position at this stage of the campaign.
"It's the same in every country. Every country always has at the beginning of the season a surprise," he said.
"Zwolle - a club in the Netherlands - was leading the competition in the Dutch league for six or eight matches. It’s always like that.
"But as I have said, what always matters is the end of May. That counts. Now it’s not so important.
"They have a very good squad. I know a lot of [Southampton's] players and a lot of players were offered to us."
