Fabrizio Romano reports the contact remains preliminary rather than advanced, with AS Roma showing no appetite to sell and no formal bid yet tabled for the 25-year-old.
Chelsea's interest reflects the growing likelihood that Fernández completes a move to Manchester City before Tuesday's deadline, leaving the club needing fresh options in central midfield regardless of how that particular saga resolves.
Koné has established himself as a mainstay in the Italian capital since his 2024 move from Borussia Mönchengladbach, starting 28 of his 29 Serie A appearances last season as Roma finished third, contributing two goals and three assists across more than 2,400 league minutes.
His stock has risen just as sharply on the international stage: capped for the first time against Italy in September 2024, he now has 18 France appearances to his name and started four matches at this summer's World Cup, having also started Roma's opening league fixture of the new campaign.
With Fernández's future still unresolved amid Manchester City's interest, Chelsea appear to be running parallel checks across the market rather than committing to any single target.
Camara remains on the list despite Monaco's public insistence they intend to keep him, and Koné's addition suggests the club are prepared to explore several routes simultaneously.
Roma's reluctance to engage seriously reflects the central role Koné has come to play under Gian Piero Gasperini, whose side start the season with genuine ambitions of competing near the top of Serie A.
Losing a player of his profile this late in the window, with little realistic prospect of finding an adequate replacement before Tuesday, would leave a significant gap in midfield - a factor that could make Chelsea's task considerably harder than simply agreeing a fee.