The Southampton manager wants the midfielder to decide his future sooner rather than later.
The Dutchman is keen for the midfielder to decide his next move before the end of the January transfer window to avoid it becoming an issue further down the line.
The 25-year-old is out of contract at the end of the season, but the club have offered him a new deal to extend his stay on the south coast.
"I spoke to Jack two weeks ago and we hope he will stay with us," Koeman told BBC Radio Solent.
"But we would like to know very soon, so then we know what will happen with Jack for next season."
Cork joined the Saints from Chelsea for around £750,000 in July 2011, following a successful loan spell at the club. He has since made more than 150 appearances for the club, including appearing in every game in the Championship season which saw them return to the Premier League.
He has made just six appearances in the league under new manager Koeman, with his most recent coming in last weekend’s 2-1 win over Newcastle, his comeback from an ankle ligament injury.
That injury had seen him miss six weeks over the Christmas period, and it now remains to be seen whether he will decide to stay with the club or look for a move elsewhere.