Cork, 25, has started just five Premier League games this season, playing as a backup to the likes of Morgan Schneiderlin and Victor Wanyama in the Southampton midfield.
With Cork's contract set to expire at the end of this season, his future at St. Mary's Stadium has been in doubt for some time having stalled over a new contract on the South Coast.
Now, the BBC suggest that Swansea are in talks with the former Chelsea midfielder over a move to South Wales before Monday's transfer deadline.
The Swans have plenty of money to spend this month having sold star striker Wilfried Bony to Manchester City for £28m, and manager Garry Monk has already added Tottenham right back Kyle Naughton to his squad.
It now appears that Monk wants to add a midfielder to his ranks, despite already having the likes of Leon Britton, Ki Sung-Yueng and Jonjo Shelvey at his disposal.
Cork is a more physical presence than Swansea's existing midfielders, and could now finally complete a move to the Liberty Stadium - 12 months since Southampton rejected a £3m bid from the Swans for Cork.