Seri has not joined his Nice team-mates for their pre-season training camp, leading to suggestions that he could be set to finalise a move away.
The midfielder had been linked with a move to a number of Premier League clubs as well as Barcelona and Borussia Dortmund but it appears that newly promoted Fulham may have won the race for his services.
Fulham won plenty of admirers with the attractive attacking football last season and won promotion to the Premier League thanks to a Play-off Final victory over Aston Villa at Wembley Stadium.
The Daily Star is quoting French journalist Nabil Djellit as claiming that the midfielder was in London over the weekend as he continues to be touted as a possible target for both Chelsea FC and Arsenal.
The report says Seri is now waiting for the two clubs to agree the deal but Chelsea will have to hurry if they want to keep things simple by triggering the midfielders £35m release clause as it expires soon.
West Ham, Spurs and even Fulham have also shown interest in signing Seri, though Stamford Bridge is believed to be his preferred destination.
Chelsea are now in a predicament with their delay in recruiting Maurizio Sarri as their next head coach but they remain optimistic that a deal could be struck before they face Perth Glory in their opening pre-season game.