The 25-year-old midfielder arrived at St Mary's in 2011 and has played an important role in the club's rise from Championship to Champions League contenders.
Cork, who also spent a loan spell with Saints in 2008, could now, though, be set for an exit with his contract expiring in the summer.
The club are still waiting to hear whether the Chelsea academy graduate wants to sign a new deal and manager Koeman appears to be getting impatient.
"No, we are still waiting about the answer of Jack and we like to know it next week," the Saints boss said.
"Of course (it is important to know his plans) because that makes plans for the future, for next season.
"If Jack likes to stay I'm very happy, if he likes to move then that means that we need another midfield player."
Cork has been in and out of the side this season due to injury, as well as the fine form of team-mates like Morgan Schneiderlin and Victor Wanyama.
A recent report suggested he would delay a decision on his future to later in the season, having belatedly been offered a new deal.
The popular midfielder's departure would be a blow to Southampton fans, who, by contrast, would celebrate the exit of Dani Osvaldo.
The club's record signing is persona non grata at St Mary's after head-butting captain Jose Fonte this time last year, leading to loan spells at Juventus and now Inter Milan.
There are suggestions the Nerazzurri are contemplating severing ties with the Italy international, who is reportedly attracting interest from QPR, amongst others.
"No (I don't know what Osvaldo's situation is)," Koeman said.
"Only what I can say is that he is for one season on loan and we expect it is until the end of the season, and after that we have to speak about that."
Osvaldo certainly has the qualities to improve Southampton's striking options, if not the personality.
Buying an additional frontman would bolster their top-four hopes, but Koeman, speaking ahead of Saturday's FA Cup fourth-round tie with Crystal Palace, seems happy with his options.
He has a number of key men returning in the coming weeks and the Dutchman rejected talk of a move for John Guidetti, who he worked with during their time at Feyenoord.
"There is nothing serious in that," Koeman said of speculation about the Manchester City striker, currently on loan at Celtic.
"Okay, maybe about that I worked with Guidetti one season with Feyenoord, but at the moment we are not looking for another attacker."
Source : PA
Source: PA