Chelsea confirmed on Monday that Terry - who has been at the club since the age of 14 - will bring to an end his 22-year spell at Stamford Bridge at the end of the season.
Terry came close to leaving Chelsea last season, but ended up penning a one-year extension to his deal in May following the appointment of Conte.
The Italian manager has favoured using Gary Cahill, David Luiz and Cesar Azpilicueta in a three-man defence this term, but emphasised on several occasions how important Terry has been in maintaining the team spirit within the camp.
Despite his lack of game time for the leaders this season, Terry revealed he held "positive" talks with Chelsea about staying on at the club but opted to depart this summer in order to prolong his playing career.
"After 22 years there is so much to say and so many people to thank at this great football club," he told Chelsea's official website.
"From coaches, team-mates and staff to the fans who have given me so much support down the years, I can’t thank you enough. There will be opportunities for me to speak further about this over the coming weeks.
Everybody at Stamford Bridge would like to express our enormous gratitude to John and wish him the very best of luck for the future. pic.twitter.com/Yo9NrXNBMs— Chelsea FC (@ChelseaFC) April 17, 2017
"I will decide on my future in due course, but for now I am committed to helping the team achieve success this season.
"The club and I have always had a fantastic relationship, which will continue beyond my playing days. We had some really positive talks but with everything taken into careful consideration I have decided it's the right time for me to leave.
"I’ve always been conscious that I depart at the right time, in the right way, and I feel that the end of this season is the right time for the club and I."
Terry has previously admitted he would consider a move into coaching - with the 35-year-old having spent the season obtaining his badges and learning Portuguese, which could point to a desire to gain experience on the continent.
But the defender revealed he still wants to continue playing football next season, but insisted his first priority remains winning the Premier League and FA Cup with Chelsea in the coming month.
"I feel I still have plenty to offer on the pitch but understand that opportunities here at Chelsea will be limited for me," he said.
"I’m eager to carry on playing and so will be looking to continue with a new challenge. I will of course always be a Blue and am desperate to end my final season as a Chelsea player with more silverware."
He added: "Finally, words cannot describe the love I have for our football club and our amazing Chelsea fans. I would like to thank each and every one of you from the bottom of my heart for the unbelievable support you've shown me over the years.
"You mean the world to me and every trophy we've won during my time at the club we've won together. I will never forget the incredible journey we've been on. Thank you so much for everything."