Terry is out of contract at the end of this season and in January said he would not be extending his stay.
The Blues insisted an offer may be forthcoming, and it arrived this week.
A Chelsea spokesperson said: “(Director) Marina Granovskaia and (chairman) Bruce Buck met with John and his agent this week and offered him a one-year contract extension.
“With it coming so late in the season, this is a big decision for John and his family and it is something that they are now considering.”
The 35-year-old defender has made 703 appearances since his debut in October 1998.
John Terry: Only 2 players to start more than 100 PL games in the last 5 seasons have better pass accuracy than Terry (90.2%) #cfc— WhoScored.com (@WhoScored) May 13, 2016
Chelsea, who can finish ninth at best, play Leicester, their successors as champions, on Sunday at Stamford Bridge, but Terry is suspended following his dismissal at Sunderland.
Chelsea sought to draw a line on the matter with the statement, allowing Terry to consider the offer and interim boss Guus Hiddink to focus on his final media conference ahead of his last game against Leicester.
Antonio Conte will arrive as head coach at the conclusion of Italy’s involvement in this summer’s European Championship.
“It’s good to have big players in a club who are big personalities,” Hiddink said.
“I’m always in favour, generally, that players, ex-players – I’m not talking about John now – are having a lot of influence in the culture of the club.
“Maybe, I don’t know when, in the future, it’s also for John.”
Terry joined Chelsea as a 14-year-old and Hiddink lauded his longevity, having seen a recent photograph of a teenage Terry.
“Twenty, 21 years at the club is huge,” Hiddink added.
“That’s why they become legends. You must cherish those legends for the future as well.”
It will be farewell from Hiddink and a return for Leicester boss Claudio Ranieri on Sunday.
Ranieri was sacked by Chelsea in 2004 and was succeeded by Jose Mourinho.
John Terry's Premier League season by numbers:— Squawka Football (@Squawka) May 13, 2016
90% pass accuracy
Still got it. https://t.co/dLOAoHItI0
Mourinho’s second spell in charge of the club ended with the sack in December after a woeful start to Chelsea’s season.
Hiddink, who won the 2009 FA Cup during his first caretaker spell, has brought stability to the west Londoners but still has only one home win in the Premier League.
The Blues drew 1-1 at Liverpool on Wednesday and are 10 points behind the Reds, who are one position above them in eighth.
Chelsea could finish 10th, if Stoke beat West Ham, but are unlikely to drop to 11th as Swansea, who play Manchester City, are three points behind and have a far inferior goal difference.
“When they asked me (to take over on an interim basis), I didn’t think much to say yes or no, because of my affection for the club,” Hiddink said.
It is announced in the press conference that John Terry has been offered a one-year contract extension. John is considering the offer. #CFC— Chelsea FC (@ChelseaFC) May 13, 2016
“I was very pleased I could do it and was asked. Of course, I would’ve loved, come the escape from the relegation zone, I would’ve loved to go to the FA Cup final, to go to at least one or two more rounds of the Champions League, but we could not fulfil that.
“In general I’m happy with this last half-year. I did it with a lot of joy.
“For Chelsea standards it has been a season with deception. We managed to come back and we’re happy to, but it’s not enough for Chelsea standards.
“It’s now relatively quiet in the club. We have to go on and be very competitive next season.”
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