The 53 member associations of UEFA also called for an age limit of 72 for all elected FIFA officials, including members of the executive committee.
The FIFA president currently serves an unlimited number of four-year terms with 76-year-old Sepp Blatter, the present incumbent, in his fourth mandate.
FIFA is due to discuss and vote on proposed changes to its statues at its Congress in Mauritius in May in a bid to make it more transparent and accountable following string of corruption scandals in 2010 and 2011.
Three members of its 24-man executive committee were sanctioned for corruption, including former presidential candidate Mohamed Bin Hammam who was banned for life, and another two resigned amid allegations of wrongdoing.
Europe holds 53 of the 209 votes and secretary general Gianni Infantino said they would vote as a block at the Congress.
Infantino said UEFA's proposal was based on the International Olympic Association (IOC) which had adopted a similar of model of an initial eight-year term with a further four years for its president.
Europe's proposal differs from a FIFA working group which has suggested a limit of two four-year terms for future presidents.
UEFA proposed that members of FIFA's executive committee members should continue to be appointed by their respective federations as they are today and that integrity checks should be conducted at regional level.
"These should be based on minimum agreed criteria. For example, a person who has been banned by a sporting disciplinary body for corruption, match-fixing or violence should not be eligible to became a FIFA executive committee member," Infantino said.
Other proposals included clearer and transparent rules on the process for proposing changes to sport's rules, which are controlled by the International Football Association Board (IFAB).