Terry was sacked by FA chairman David Bernstein on Friday morning after the Chelsea defender saw his trial for alleged racial abuse adjourned until July.
Steven Gerrard has emerged as the favourite to succeed Terry as England skipper, while Frank Lampard is also likely to a strong candidate.
Manchester United centre-back Rio Ferdinand has ruled himself out of the running following his past experiences in the role, while Tottenham midfield Scott Parker is seen as an outside bet to take the armband.
Hart, meanwhile, is seen to be a strong contender as he is one of the only players to be able to boast a guaranteed place in Fabio Capello's first-choice XI.
And, although the 24-year-old is flattered to be linked with the position, he does not expect to be in contention when Capello makes his decision on the situation.
"I'm flattered to be linked with it, but I think anyone who's having a consistent run in the England side is being linked with it," he said, according to skysports.com.
"There are stronger candidates than myself.
"I think there's a long way to go [before the captaincy].
"I've got a really important season, games for Man City and hopefully a couple for England coming up.
"That's all I'm concentrating on at the moment. As long as I'm playing, I don't care who's captain."