The 35-year-old defender is out of contract in the summer and shocked fans by saying back in January that he would leave Stamford Bridge at the end of the season.
But the Blues continued to insist that the skipper could yet be offered a deal to stay, and it arrived this week.
A club representative revealed on Friday: “[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.”
But Terry admitted later on Friday that he is considering whether or not to accept the offer.
He wrote on Instagram: “The contract extension the club has offered me is a different role and I hope everyone will understand I want to take the time to consider it carefully before making a decision.”
Italian coach Antonio Conte will take charge of the Blues for next season, and outgoing manager Guus Hiddink said in a recent press conference that clubs need characters like Terry.
“It’s good to have big players in a club who are big personalities,” Hiddink said of the former England captain, who joined Chelsea as a 14-year-old.
“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.
“Being at the club for 20/21 years is huge. That’s why they become legends.
“You must cherish those legends for the future as well.”