The 25-year-old former Chelsea youngster is out of contract in the summer and Swansea boss Garry Monk hopes to sign Cork for a reduced price, although Press Association Sport understands no fee has yet been agreed between the two Barclays Premier League clubs.
Southampton boss Ronald Koeman has told Cork that he must decide his future by the end of this week with the transfer deadline looming on Monday.
"I spoke to Jack two weeks ago and we hope he will stay with us," Koeman told BBC Radio Solent last weekend.
"But we would like to know very soon, so then we know what will happen with Jack for next season."
Cork has played nearly 130 games since joining Southampton in 2011 but he has started only five Premier League matches under Koeman during a season where he has suffered injury and been behind Morgan Schneiderlin, Victor Wanyama and Steve Davis in the midfield pecking order at St Mary's.
He made his comeback after six weeks out with an ankle ligament injury in Southampton's 2-1 win at Newcastle on January 17 and started last weekend's FA Cup defeat to Crystal Palace.
Monk wants to make Cork his fourth signing of the January transfer window after spending £5million on Tottenham defender Kyle Naughton and landing Exeter youngster Matt Grimes as well as taking Benfica striker Nelson Oliveira on loan.
Cork's father Alan, the former Wimbledon striker, is a former Swansea manager having been in charge of the club during the 1997-98 season.
Source : PA
Source: PA