Infantino is under pressure after being forced to abandon plans to seek private investment in a company to run FIFA tournaments, including the men's and women's World Cups.
Three continental confederations accused Infantino of "deception" in an open letter last week over the manner in which the plan was devised and presented, with European football's governing body UEFA still intent on boycotting FIFA events having "lost confidence" in Infantino's leadership.
The Swiss does still retain support, notably in Africa and South America, but now fans' organisations have called on him to step down, with a petition being launched on the change.org website.
"The shameful attempt by one individual – the FIFA president himself – to sell off the World Cup to private investors exposed a betrayal football will never forget," a statement on behalf of the fan organisations said.
"Infantino's plan was stitched together in secret. Negotiated behind closed doors. Hidden from FIFA's own members and from the millions of people whose passion makes football what it is."
Individual fans are being encouraged to sign the petition, which already has support across all six FIFA confederations spanning continental and national fan groups.
The private investor plan is seen by the fan organisations as the final straw, on top of FIFA's involvement in discussions around the European Super League and its ticket pricing policy at this summer's men's World Cup.
"A president who repeatedly betrays football and makes a mockery of our game has lost the right to lead it," the statement continued.
"For the credibility of FIFA – and for the future of the game – he must resign."
The petition also calls for reform of FIFA itself, including the formal involvement of supporters and other key football stakeholders in decisions that shape the future of the game.
Ronan Evain, the executive director of Football Supporters Europe (FSE) which is a signatory to the petition, told the Press Association: "The departure of Gianni Infantino is a first and a must, but that can only have a positive impact if it's followed by some reform of the office and of governance.
"If a man can concentrate so many powers then the governance of FIFA is not fit for purpose.
"(FIFA) needs to start from scratch. All the reforms from 2016 have been erased or made irrelevant by Infantino. I think we need to be as open-minded as possible on the future of football governance and there are other models outside of football, outside of sport, that need to be looked into – the idea of rotating presidencies seems interesting."
FIFA has been approached for comment.
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