The 2009/10 Premier League champions announced this month that they had made a proposal to buy back the freehold land at Stamford Bridge from Chelsea Pitch Owners (CPO).
Chelsea said this did not mean the club would definitely be relocating but added that buying back the freehold removed a potential hurdle should a suitable alternative site become available in the future.
England skipper Terry, 30, is president of CPO but he said he was excited at the possibility of moving to a new ground.
"I can understand people being a bit wary of this proposal but they've seen what the owner has done in a short space of time and we have to keep faith with him," Terry told the club's website.
"He will do the right thing now and he will still be doing the right thing in 10 years' time. He realises that without the fans Chelsea is nothing.
"I am one of them, my kids are too, and I want to be part of that for the next 40 to 50 years. What excites me is the thought of my kids... being able to grow up watching the team play in a big stadium."
Stamford Bridge has a capacity of 41,000 but Terry said Abramovich, who bought the club in 2003, was keen to have a bigger stadium to match Chelsea's growing aspirations.
"I wouldn't be coming out speaking publicly and backing the club if I didn't believe in it," said the centre-half. "I could quite easily say nothing and let the club do their thing but I've been here so long and I've spoken to the owner on numerous occasions.
"He is Chelsea mad and wants us to be in with the Barcelonas, the Manchester Uniteds, and we know that doesn't happen overnight. We can grow if we are successful on the pitch and we've got the capacity to expand.
"Obviously, I have great memories of Stamford Bridge. I made my debut at Stamford Bridge and have spent my whole career at the one club," said Terry.
"If the club does end up moving then Stamford Bridge will never be forgotten by myself or the fans. But for us to go further we might need to move and we have to trust our owner."
