Football's governing body has repeatedly come under fire over the years, amid accusations of corruption. At the heart of these attacks is the football association's president Sepp Blatter.
Plans to hold the 2018 and 2022 World Cup in Russia and Qatar respectively have been heavily criticised, with many observers saying the successful bids were the result of bribery.
In 2012, Fifa named US attorney Michael Garcia as chair of the investigative branch of its ethics committee, to look into the accusations concerning Russia and Qatar. At the end of last year, a summary of the report was released by German judge Hans-Joachim Eckert, which cleared the two countries of any wrongdoing and failed to protect the identities of whistleblowers. Garcia subsequently resigned in protest.
Welcoming members of the public and fellow MEPs to a summit on a coalition for Fifa reform, Emma McClarkin, a vice-president of parliament's new sports intergroup, said, "sport should be governing itself but we have a duty as external stakeholders and fans to step in when that doesn't seem to be working".
Belgian MEP Ivo Belet explained, "we fully respect the principle of self-regulation for football governing bodies but on one condition - that good governance within the sporting bodies is guaranteed".
Harold Mayne-Nicholls, who led Fifa's technical inspection team for the 2018 and 2022 World Cup bids, suggested that Fifa's presidency should be limited, "with no more than one re-election". Sepp Blatter has held his position for over 16 years and has been re-elected three times.
Perhaps anxious not to ruffle any feathers, Jérôme Champagne, who is running for the Fifa presidency, stressed that "the democratic presumption of innocence should be respected, including in Qatar".
"Sport should be governing itself but we have a duty as external stakeholders and fans to step in when that doesn't seem to be working" - Emma McClarkin
In addition to allegations of corruption, Qatar has also been hit with accusations that it is using slave labour to build its stadiums in preparation for the World Cup. The international trade union confederation has predicted that 4000 workers would die building the new facilities.
Champagne said, "unfortunately a World Cup means revamping and building new stadiums, and accidents can happen", but warned that "we cannot celebrate football and we cannot rejoice knowing the conditions in Qatar".
Noting that Fifa's problems were "a collective responsibility" and that it was "unfair to blame Blatter", he promised to "continue what has been done correctly in the past 40 years, such as the development programmes" if elected.
However, it should be noted that these development programmes have been suspected as being a means of discreetly circulating funds between Fifa officials.
Bonita Mersiades, who was a senior executive of the Australian World Cup bid and one of the whistleblowers exposed in Fifa's report summary, said that "funding for development purposes has long been the way Fifa members have sought support in one way or another".
She explained that when Australia was vying to host the competition, €2.8m were donated to fund various sports development projects, as requested by a member of Fifa's executive committee.
Marc Tarabella, who is chair of parliament's new sport intergroup, called for the football association's elections to be "the beginning of more transparency around decision making and financial transactions".
According to the Belgian MEP, "more builders will die in Qatar than players will play in the World Cup".
Concluding the session, Emma McClarkin said, "this is just the beginning - it goes beyond the elections".