FIFA has told member associations they will each receive $40 million if they back a plan to sell stakes in major competitions -- but they must sign up by September 19.
The sport's global governing body on Tuesday announced proposals to set up a commercial subsidiary to run events, including the World Cup, sparking a furious backlash.
European football's ruling body UEFA said the World Cup was not an "asset to trade", while the Asian Football Confederation said it had not been consulted.
EU sports chief Glenn Micallef told FIFA: "Hands off our game."
According to The Times on Wednesday, FIFA president Gianni Infantino told members that a funding package totalling $10 billion would become available if they signed up.
Infantino said in a letter seen by the British newspaper that the decision was up to individual associations.
"Should you wish to proceed, this $10 billion package will become available as of January 1, 2027, ushering in the next phase of our journey together," he wrote.
"Should you wish to retain the status quo and reject this proposal we still have our planned expansion of the Forward programme (development funds) of $2.7 billion as previously presented.
"In total for the upcoming cycle starting as of January 1, 2027, each member association will have the possibility of access up to $40 million per member association under this proposal."
The Times said sources opposed to the plan had labelled it "pure bribery".
UEFA issued a scathing fresh statement on FIFA's proposed timescale.
"Today we have learned of FIFA's deadline to associations to support their proposals or have the one-off payout offer withdrawn," it said.
"This says everything you need to know about this plan.
"But having held discussions with many stakeholders across the game, UEFA knows there is significant and growing opposition to FIFA's scheme.
"FIFA cannot continue to use our sport to enrich themselves and their friends. We can grow the game correctly. It's time to prioritise associations, clubs, leagues, players and fans."