The striker is the highest profile member of the group whose existing deals expire at the end of the season.
He is also the longest-serving player in the squad, making 263 appearances since joining, initially on loan, in 2008.
The 29-year-old has been their top scorer in all five seasons since and is also on the verge of becoming United’s all-time top marksman.
An impressive finish in Saturday’s 1-1 draw at Bristol Rovers, which ended an 11-game barren run, was his 104th goal for United, just four away from breaking the record.
Constable, known as Beano, is keen to agree a new deal, but little progress has been made and he knows the ball is in chairman Ian Lenagan’s court.
“My dad’s my agent and he has put a couple of calls in, but other than that we’re not too much further down that line,” he said.
“I’ve made it pretty clear I’d love to stay. It’s over to the chairman now and hopefully it can be sorted out.”
He added: “Time is ticking away and I’m desperate to get something sorted and stay at the football club.
“I’m sure in the next few weeks that will be tied up, we just need to carry on getting results on the pitch and the rest will take care of itself.”
Lenagan has made it clear in recent weeks he has no intention of repeating the end of last season, when decisions had to be made on the futures of a large chunk of the squad.
Progress is being made, with goalkeeper Ryan Clarke and youth team prospect Josh Ashby agreeing new deals in the last fortnight.
Seeing a contract run down is a new experience for Constable, but he is focused on putting the uncertainty to one side.
He added: “It’s normally been done years in advance, but at 29 it’s probably the way it will go.
“It’s not affecting me at the minute. I’m just desperate to get out there and perform the best I can.
“The more goals I score, hopefully the easier the decision will be.”
- James Constable's strike at Torquay United (at 3.05) has been short-listed for Mitre Goal of the Year