Blatter, who took over as the world governing body's chief in 1998, became embroiled in controversy over the weekend when he suggested that criticism of FIFA's decision to award the 2022 World Cup to Qatar were founded in racism.
In a meeting with UEFA officials on Tuesday, Blatter was asked to stand down by Michael van Praag, head of the Dutch Football Association (KNVB), and David Gill, vice-chairman of the Football Association (FA).
Gill told Sky Sports News he felt Blatter should step down, after the Swiss announced in 2011 that he would do so, but since suggesting he will stand again.
"My personal view is he should have stayed with his comment back in 2011," Gill said. "I think he has done some good work over the years.
"Yes he is looking at the governance structures and what needs to be done going forward. However, you can change the structure but unless you have the right people within it, the structure is irrelevant."
Greg Dyke, chairman of the FA, suggested that the governing body in England wanted the Swiss to make way for an alternative candidate, namely UEFA chief Michel Platini.
"The FA's position is we accepted four years ago that he was going to step down and there's some surprise that he's decided to stand again," he said.
"We haven't yet taken a decision, but we would like to see an election, competition and different candidates.
"I think the view probably is it's time for a change, yes. He's made it clear he's going to stand again and I suspect he thinks he's got the votes to win.
"The most credible candidate is Michel Platini and he will decide in the next few weeks whether or not he will stand. If he doesn't, we'll see who does, hopefully there'll be other candidates."
Van Praag - tipped as a potential successor to Platini, should the Frenchman replace Blatter - explained the details of Tuesday's meeting.
"I took what Sepp Blatter said to us as an official statement that he would be running for re-election so I took the floor," the Dutchman explained.
"I told him it was not a personal attack, because I respect him and know him and like him as a person, but FIFA has an executive president and that means he has the final responsibility for what is happening in FIFA.
"If you look at the way people think about them at the moment, they have a very ugly reputation.He bears the final responsibility for that."
