The 25-year-old took all the plaudits after a game-saving display against Borussia Dortmund in the Champions League last Wednesday, in which he made a number of fantastic saves.
However, the former Shrewsbury Town man suggested people can be quick to hail goalkeepers after one good display, as well as it working the other way, but is still delighted with the praise.
"It's nice people say nice things but things are easily said," he told reporters.
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"It was great and a real special honour for some special people to say some nice things but we move on.
"You've got to remain level-headed. If you don't, it's a big slap when you do something wrong."
The City keeper revealed that he does not believe he is the best in the world, but that he is aiming to get there one day.
"I'm still trying to improve," he admitted. "I hate to say it but I am getting a bit older now so I have to get in a bit earlier and prepare my body a bit differently.
"I just know every day I want to be the guy you can rely on. I want to be the best."
England manager Roy Hodgson is likely to be without Frank Lampard who is an injury doubt, and the suspended Steven Gerrard against San Marino, and many have touted Hart as a possible stand-in captain.
But the Three Lions' number one stated that he has not even considered himself as an option for the armband, indicating his belief that Rooney's experience means he is the right choice.
He added: "I've not thought about it at all. How the captain is chosen is the manager's decision.
"I've got a big responsibility to keep my place in the team for club and country.
"If I do get the nod, I'll try and give as much as I can in terms of helping the team.
"But I think it's between Wayne Rooney and Wayne Rooney if we are going to go on experience and how captains have been chosen in the past."