The Welsh internationalist, who captained Celtic yesterday, still has more than a year on his current deal to run, but is set to be rewarded.
Speaking after yesterday's 5-0 win, manager Neil Lennon revealed: "He is very important here and has proved that.
"He is talking about staying on and I would like him to do that as he is good for us. We have had preliminary talks and we'll go back to his representatives."
Meanwhile, Lennon was delighted with the emphatic nature of his team's victory over Dundee, despite a few sloppy first-half episodes.
Efe Ambrose was subbed just three minutes before the break having filled in at right-back and Lennon felt the Nigerian just wasn't at the races.
"Efe played there for his country at the African Nations but, on reflection, he looked a bit fatigued," said the Hoops boss. "He didn't play his best football but he knows that."
With a number of games looming now, Lennon will look to utilise the full squad as the club try to reach the 92-point mark for the season.
Hitting that target requires an unblemished return between now and the end of the season.
"We have a lot of games coming up," he said. "It was a Jekyll and Hyde performance yesterday. In the first half we weren't great, but in the second half we were excellent.
"I am pleased with Dylan McGeouch. It was a good goal and a good performance and that is two good 90 minutes' he has put in for the team.
"Gary [Hooper] is up at 24 goals for the season and it was a good all-round performance, although I am sure the turning point was the sending-off.
Asked who would take the club's next penalty after James Forrest scored from the spot, Lennon said: "I'll decide before the next game. It will be interesting to see who gets the ball first. Kris [Commons] was out with a virus and we'll need to see how he is going to be for Wednesday."