Michael van Praag, the president of the Dutch FA and Greg Dyke, the chairman of the English FA, openly challenged Blatter after he addressed the European delegates at their national association meeting before the FIFA Congress which begins on Tuesday evening.
After Blatter had told the assembled UEFA delegates he was seriously considering standing for a fifth term, Van Praag addressed the Swiss head of world football, the Dutchman later told reporters.
"Mr Blatter, this is nothing personal but if you look at FIFA's reputation over the last seven or eight years, it is being linked to all kinds of corruption and all kinds of old boys' networks things," Van Praag said he had told Blatter in the meeting from which reporters were barred.
"FIFA has an executive president and you are not making things easy for yourself and I do not think you are the man for the job any longer.
"I told him 'I like you very much ... this is nothing personal. But you are now saying that Qatar was the wrong choice (for the 2022 World Cup), but you are not blaming yourself you are blaming your executive committee.
"Yesterday you said something about racism against Qatar and people are not taking you seriously any more. This is not good for FIFA and it is not good for the game.
"Yes you are leading the reforms at the moment, but all these problems occurred in the period before the reforms and you were still president and you responsible and I believe you should not run any more."
On Monday, Blatter branded some of the criticism of the Qatar World Cup award for 2022 as racist and launched a scathing attack on those he said were "plotting to destroy" FIFA.
Blatter's comment came in the wake of a series of fresh allegations made by Britain's Sunday Times newspaper about the award of the Cup to Qatar and rumblings from sponsors who are unhappy with the current trouble FIFA is facing.
Van Praag said the furore had "nothing to do with racism" while Dyke branded Blatter's comments as "offensive".
"I said 'could I say that I regard the comments you made yesterday about the allegations in the British media in which you described them as racist is totally unacceptable'," Dyke said.
"The allegations being made have nothing to do with racism, they are allegations about corruption within FIFA."
Van Praag said that Blatter had misunderstood what he said when he addressed him.
"He thought I was calling for him to resign today. I wasn't. That was not what I meant to say at all, but that's what he thought I said and he (Blatter) said he would not resign he had another year of his mandate to run."
Asked if UEFA had an alternative candidate to Blatter, Van Praag added: "No. (UEFA chief) Michel Platini has discussed this with many members but has not made up his mind yet.
"Since that is the case UEFA does not have another candidate."
UEFA took no resolution against Blatter, but the mood of the meeting had been "tense" according to a UEFA spokesman who said Blatter received only polite applause at the end.
After the meeting former UEFA president Lennart Johansson, who lost a FIFA presidential election to Blatter in 1998 told reporters: "It is time he went. He has done some good things for football and FIFA but 16 years is enough."
Asked if UEFA were the only confederation who would like to see him go, UEFA executive committee member Karen Espelund of Norway said: "We will see, he has had some clear messages today and there is still a year to go."
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