The 35-year-old defender has made 703 appearances since his debut in October 1998, but was suspended for the season-concluding 1-1 draw with Premier League champions Leicester City.
Terry, whose current contract expires on 30 June, in January said he would be leaving Chelsea at the end of the season. The club insisted an offer may be forthcoming and it arrived last week.
The defender said it was for a 'different role', hinting at reduced terms and without guarantees of a first-team place under incoming head coach Antonio Conte.
Terry addressed supporters after the contest, with the fans serenading the man they describe as 'Captain, Leader, Legend' on his potential farewell. "John Terry, we want you to stay," the supporters sang.
"We all want the same thing," said Terry, who was thrown aloft by his team-mates on a lap of appreciation after the match and his speech.
"I've said for a very long time I want to be here for the rest of my career. I want to finish my career here.
"We'll have a few days and we'll be speaking to the club. Listen, I want to stay. The club knows that, the fans know that."
While owner Roman Abramovich applauded his captain, the comments may suggest Terry wishes to negotiate the deal which has been offered. He is reportedly fielding offers from elsewhere, including cash-rich China.
Home supporters sang about Terry throughout, but banners were unfurled midway through the first half when 26 minutes were on the clock - Terry's shirt number.
He acknowledged the adulation by beating the left side of his chest, over his heart and the club crest which adorned his t-shirt.
Terry endeared himself yet further to supporters when congratulating Leicester and Claudio Ranieri, the Foxes boss and his former manager, on their title success.
"I'm delighted Leicester have won it and I'm just glad Tottenham haven't," Terry added.
Twelve months ago Terry lifted the trophy. Chelsea finished this season in 10th place.
"We are where we deserve to be in the league," Terry added.
"At times we've all accepted we've not been good enough.
"But I can assure you, we'll be back next year and we'll be fighting for this title.
"Chelsea is my life, just like it is yours. You've been a big part of the best moments in my life in this stadium.
"From myself and all the players I'd like to say a big thank you."
He also thanked Guus Hiddink, whose second spell as Chelsea interim boss came to a close after the match.
"He's picked us up," Terry said.
Hiddink was pleased Terry and Chelsea are now in contact over the new deal, but declined to speculate on the future.
The Dutchman said: "It's good when both parties start to talk.
"Personally I hope they manage to come to a solution.
"He's still physically fit and mentally very fit, so he can contribute a lot wherever he wants to play."
Ranieri paid tribute to Terry.
The Leicester boss, who managed Terry in his formative years as a professional, said: "John Terry is as my son. I believed in him when he was young.
"I put him in the team when there were a couple of champions of the world and European champions, (Marcel) Desailly and (Frank) Leboeuf.
"He played, because for me it was better to put him in the squad.
"He made a fantastic career. I hope he can continue."
