Mayweather, 39, who retired from boxing in 2015, said on Wednesday talks about a fight had taken place and claimed he had offered the Irishman $15m.
But UFC president White said: "He hasn't made offers to anybody.
"Floyd, here's a real offer - and I'm the guy who can make the offer - we'll pay you $25m and Conor $25m."
Speaking to a Fox radio broadcast, White said that the two would be likely to meet in a boxing ring, not the UFC octagon.
Each fighter would also get a cut of pay-per-view revenue, he said.
"If they're going to box, Floyd looks at this as an easy boxing match for him," White added. "Conor McGregor is his money fight."
Mayweather, who retired with a perfect 49-0 record, told ESPN's First Take programme that a McGregor bout was the only thing that could tempt him to end his exile from the ring.
McGregor, 28, became the UFC's first dual-division champion in November and has previously challenged Mayweather to a fight under mixed martial arts rules.
His boxing licence has been granted by the California State Athletic Commission, allowing him to box in the US state.
However, he is under contract with the UFC and any potential fight against Mayweather would have to involve a co-promotion or the UFC's blessing.
Ultimate Fighting Championship boss Dana White says he will pay Floyd Mayweather and UFC champion Conor McGregor $25m (£20.5m) each to fight.