The 29-year-old, nicknamed Beano, is the highest-profile member of the squad whose existing deals expire this summer.
A haul of 11 goals so far this season has put him on course to become their top scorer for the sixth successive season, and leaves him only three away from their all-time record.
Constable, who joined in 2008, is the longest-serving player in the squad and last month admitted he was “desperate” to stay.
In the absence of a permanent manager, Ian Lenagan is handling decisions on contracts, along with caretaker boss Mickey Lewis.
The U’s chairman has revealed he is scheduled to meet Constable’s agent, who is his father, Stuart, today and has also held talks with at least two other players.
Lenagan said: “Mickey and I are heavily involved in contract discussions with a number of players.
“We have had a lot of discussions with Josh Ruffels and Tom Newey and a number of other players.
“I’m not excluding anybody by mentioning those three.”
Lenagan has been clear he does not want a repeat of the end to last season, when the club had to make decisions on most of the first-team squad.
Goalkeeper Ryan Clarke and youth team prospect Josh Ashby are the only players to sign new deals so far, but more appear to be in the pipeline.
Lenagan said: “It’s an ongoing thing and I’ve had years of experience of doing that with players in contract situations.
“I’m quite comfortable about where we are for the end of the season.”
United are among the promotion contenders, but there is no intention of waiting to see which division they will be in for the next campaign.
He added: “It’s a factor, but many of the players we are talking about – and this was Chris Wilder’s view as well as Mickey Lewis’s – is they are capable of playing in League One.
“There’s no panic about whether we are in League One or Two, but we obviously want to be in League One.”
- JAMES Constable’s volley at Torquay United failed to make the final shortlist of five for the Mitre Goal of the Year at the Football League Awards later this month.