The England midfielder has just over a year left on his current deal and several months ago Koeman set a deadline of the end of the season for the 23-year-old to re-sign or be sold.
Koeman has reiterated the club needs to have an answer by next weekend — when Everton finish their Premier League campaign at Arsenal.
However, the Dutchman is by no means confident there will be a successful outcome and has already started looking at replacements in offensive positions.
“It is still one week. Time for him to decide,” said Koeman ahead of tonight’s visit of Watford.
“The board has been working a long time to sign Ross for him to stay for a longer period but it is up to the player.
“But we don’t wait till August. Next weekend we need an answer.
“If we go on holiday we need to know. Either he accepts the contract or we sell the player.
“But if you need so much time then you have doubts.
“I like to work with players who like to stay. We are already trying to get players in for next season in the front positions.
“It is up to the player and he needs to make a decision.”
One player who is due to sign a new contract is Kevin Mirallas. The other player whose future remains in the balance is striker Romelu Lukaku, who has told the club he will not be signing an extension to his existing contract.
The Belgium international, who is the Premier League’s top scorer with 24 this season, has two years left on his deal but has been linked with a move back to Chelsea and also a switch to Manchester United.
Lukaku won both player of the season and the players’ player award on Tuesday and, in his acceptance speech, he thanked Koeman for helping him improve after admitting he was “not always an easy guy”.
“He is not so difficult. He knows what he needs to improve,” said Koeman, who had no update on the striker’s future.
“I think he improved in several aspects in his play but I think he can do better in several aspects of pressing, in working for the team.
“We know his big quality is in finishing — I can’t make him better in that but I can make the player better in other aspects.
“If you see the performances of Rom overall this season they are better than they were and that is what you like to do as a manager, to develop players and make the team stronger.
“He has scored more goals than he scored last season so that is really positive.”
Meanwhile, Watford manager Walter Mazzarri admits Tom Cleverley will be missed against Everton, but is confident his squad have enough strength in depth to get the job done.
Cleverley returned for what was his second loan spell with the Hornets in January, and it is a move which is set to be made permanent in the summer. The 27-year-old, though, is not eligible to face the Toffees at Goodison Park.
Swiss international Valon Behrami is expected to be drafted into the team alongside Abdoulaye Doucoure.
“He (Cleverley) has been a very good player and we will miss him tomorrow, but the positive is he will be fresh against Chelsea (next week),” Mazzarri said yesterday.
Watford have been linked with a summer move for ex-Arsenal defender Thomas Vermaelen, who is currently on loan at Roma from Barcelona. “He (Vermaelen) is a very good player, but let’s not talk about the transfer market,” the Watford boss said.
“I don’t want to talk about it. We still have three very difficult games left.
“At the end of the season we can talk about it. At the moment I want my players concentrated on the next three games.”