Blatter, 78, is expected to announce his intentions to run for a fifth term in 2015 at the FIFA congress in Sao Paulo on Wednesday.
But during an address to Europe’s top football officials on Tuesday, former Manchester United chief executive Gill, along with fellow committee member Michael van Praag, told Blatter he should not be a candidate because FIFA's reputation has been so damaged during his presidency.
Blatter and FIFA have come under increasing scrutiny in recent weeks following a sensational exposé in the Sunday Times newspaper that alleged corruption was at the heart of Qatar’s successful 2022 World Cup bid, although the organisers vehemently deny any wrongdoing.
Blatter condemned the reports on Monday, suggesting to the African and Asian football federations that they were motivated by “discrimination and racism” from the British media.
Van Praag, the Netherlands federation president, said "people link FIFA to corruption and bribery and all kinds of old boys' network,” as he and other delegates met Blatter at UEFA’s World Cup base.
And Gill told Sky Sports News:”From our perspective we were surprised that he declared his candidacy because in 2011 he was very clear that it was his final four years.
"My personal view is he should have stayed with his comment back in 2011.
"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.
"FIFA is the over-arching body in world football and needs to have a good relationship with all the confederations. What we would like to see is a full and frank open debate around the process going forward."
Van Praag also described the moment he took the floor and told Blatter he should stand down as promised at the end of his current four-year term.
"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," he said.
"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.
"After that moment there was a huge silence after I stopped speaking."