John Cross in The Daily Mirror reported that Terry is "resigned to leaving" but wants a long-term deal, Dominic Fifield claimed in The Guardian that Jose Mourinho will press for a new deal for his captain, while The Sunday Mirror's Steve Stammers said that, in line with Chelsea club policy about giving contracts to players over 30, Terry will be given just a one-year deal.
If the decision is entirely down to Jose Mourinho (as Jeremy Wilson in The Daily Telegraph suggested) then one would suspect Terry will still be at Stamford Bridge next season.
His Chelsea career looked all-but over last season, with injuries taking their toll and Rafa Benitez reticent about picking the club captain. Terry started only 14 of the 32 games he was available for under Benitez, and nine of those were in the Europa League and FA Cup, competitions that are very much lower priorities for a club such as Chelsea.
He never started more than two games in a row. Terry has a history of making rapid recoveries from injury, suggesting that he frequently played through pain, which logically one would assume would catch up with him, and it very much looked like that was the case.
However, this term, since Mourinho's return, Terry has played every minute of every Premier League game, only rested for the domestic cup games and the early Champions League tie against Basel. This has included playing three games in a week on no less than eight occasions, something you'd think was pretty unlikely for an apparently over-the-hill 33-year-old with a crumbling body.
A large part of Terry's return might have been down to the affinity he, and the likes of Frank Lampard, clearly feel for Mourinho from the manager's first spell at the club. Terry told ChelseaTV (quoted by The Daily Telegraph) in December:
The last couple of years we've lost our identity a little bit. There's some good teams in the league and when you go 3-2 up (for example) it's important you get men behind the ball and dig in deep and not give them opportunities to really feed on. I think we're doing that really well at the moment.
So should Chelsea keep Terry on? Based on his performances this season, they should. Chelsea have the best defence in the Premier League this season, something that has plenty to do with keeping four clean sheets in their last six league games, a noticeable recent tightening since they kept the same amount in the previous 18. Terry has obviously been a key part of that.
Another important factor to note is that Terry's game will not necessarily suffer for age. He has always been pretty slow, so the advancing years and declining speed won't exactly take a big chunk out of his skill set.
Terry has always been more of an organiser, a man to throw himself in front of a shot and make a last-ditch tackle, rather than keeping an opposition striker in check with pace.
Also, like when a chairman takes a decision to sack a manager, Chelsea must consider who would replace Terry if he leaves. There are no outstanding current candidates in their squad, with Gary Cahill unreliable, David Luiz perhaps a little erratic (and arguably more useful in midfield) and Branislav Ivanovic more comfortable at right-back.
As for other recruits, Chelsea's priorities in the summer lie elsewhere, most obviously in buying a top-class centre-forward, but also in replacing Ashley Cole, who definitely looks like he'll be leaving, per the Daily Mail's Sami Mokbel.
A big caveat is Terry's injury history. A body as battered as his will be unpredictable, so there is no telling at what point he might completely fall away, thus a one-year deal then a reassessment at the end of the year looks sensible.
Whether he will agree to such a short-term contract is another issue, but Chelsea would be making a big mistake if they let their captain go simply because he is getting on a bit.