The midfielder is entering the final year of his current deal.
The Daly Echo says Cork's future appeared to be in some doubt early in the season, after he lost his place in the team to summer signing Victor Wanyama.
But, after the £12.5m man suffered an injury in December, Cork took his chance, re-establishing himself alongside Morgan Schneiderlin in the defensive midfield area.
The 24-year-old's performances since his return have been so impressive that he was even name-checked by the England manager, Roy Hodgson, recently.
For breaking stories and all the great banter like us on Facebook: facebook.com/tribalfootball