United submitted an offer for Schneiderlin two weeks ago, Koeman has confirmed, but the Red Devils have failed to follow that up with a second approach for a player Saints are said to value at £25million.
Koeman said: “There’s a lot of speculation, but two weeks ago we had a bid of Manchester United, it wasn’t enough, it wasn’t serious, and no more bids till today.
“Everybody knows the situation of Morgan. I understand the player – if he can get a big move to a big one (club), I can understand it.
“I hope that if there is any business, that it will be soon because it’s best for the player, best for us, we have to prepare ourselves if we sell the player.
“No, of course I’ve had a lot of meeting with Morgan. I had a meeting last week with Morgan, everybody knows his situation. But it’s important for the player to do a good pre-season. And we will see what happens in the next few days.
“I don’t talk about money, about numbers, it’s not my business, it’s up to the club.”
Former France boss Raymond Domenech believes Schneiderlin must join a Champions League club this summer to stand any chance of representing France at next summer’s European Championship.
Koeman also confirmed Saints’ head of football development Les Reed is attempting to push through the permanent transfer of Toby Alderweireld from Atletico Madrid.
Southampton believe they have an option to sign the Belgium defender for £6.8million following a successful loan last year and say Atletico have missed a deadline for cancelling that agreement.
Tottenham are also interested, and Koeman said: “What happened with Tottenham in the middle I don’t know. Les Reed is working on that situation and I wait for more news.”
Asked if he still wanted the player Koeman said: “Of course, and the player told us the same before the summer.”