Lallana's sparkling form this season has caught the eye of Manchester United manager David Moyes.
Cork said: "When Saints went down to League One I thought he would move but he stayed and showed loyalty to the club and brought them up two divisions.
"I don't see why he wouldn't want to stay if the club can match his ambitions because he stayed while they went right down to the bottom.
"It would be nice if he did become a Matt Le Tissier type figure. I know everyone in the city thinks so highly of him and there is no reason he can't emulate that.
"I think that bad phase with the club has help him become such a good player. Now he is showing how good he is."