The club captain had looked set to leave Stamford Bridge after 18 years in the first team, announcing in January that his contract would not be renewed.
However, he spoke more recently of his desire to stay and it was revealed at Friday's media conference that the 35-year-old had met with chairman Bruce Buck and director Marina Granovskaia to discuss fresh terms.
The club has finalised its offer and said Terry was considering his options, with a new manager arriving for next season in the shape of Antonio Conte.
Terry has established himself as one of the London outfit's all-time greats, playing an integral role in domestic and European success.
He has won four top-flight titles, five FA Cups and the 2011-12 Champions League, adding a Europa League triumph the following season.
It was feared he had played his last game for the club, though, when he was sent off in Saturday's 3-2 loss to Sunderland to incur a ban for the final two matches of the season.
Despite reported interest from across the globe, it is thought Terry's preference is to stay put as Chelsea seek to put an abysmal campaign behind them.
Their title defence faltered from the outset, costing Jose Mourinho his job, and they are set for their lowest Premier League finish since 1995-96.
Monday's match against Leicester City will be interim boss Guus Hiddink's last in charge.
