The American left-back has a long-term contract at Fulham that will take him to 2028.
However football.london states that Chelsea want a left-back to replace Ian Maatsen.
The defender has been allowed to depart the club for Borussia Dortmund this winter.
Robinson would also be coming in as a replacement for Marc Cucurella, who Chelsea also want to sell.
Whether the USA international wants to make the move, given Chelsea's squad uncertainties recently, is unclear.