"We need an answer before next week", Koeman told reporters on Thursday.
The Belgium global, 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.
"Either he accepts the contract or we sell".
As reported by the Independent, Koeman said: "It is still one week".
"One, he signs that contract. We all know it will be hard".
"We don't wait till August", Koeman said.
The 34-year-old has been at the club since joining from Sheffield United for a fee of £4m in 2007 and is contracted to the Toffees until the summer of 2018.
British PM May's Conservatives open up record lead: ICM poll
One user wrote: "Time for her to leave and let Scotland prosper in a strong union". She added: "Let's take the Tory argument at face value".
"If you take so much time (in signing) you have doubts (about staying). At the moment I want my players concentrated on the next three games".
The Dutchman has repeatedly criticised his player in public over the way he is developing, and earlier this season he was struggling to get into the first team. "He knows what he needs to improve", said Koeman, who had no update on the striker's future.
"I spoke to him and we need more productivity from him".
Tottenham manager Mauricio Pochettino is renowned for managing to get the best out of talented young players and the signing of Barkley would certainly tally with the club's philosophy.
Costa has bagged 12 goals across all competitions, adding eight assists throughout the campaign, leading to him landing four awards at Wolves' end of season dinner.
"You can never say never but I am looking forward to finishing off the season well, having a break and then coming back all guns blazing and trying to keep hold of my place in the team".