The England U-21 international keeper is believed to have held initial talks with the Londoners and could complete a move before the end of the week.
Leeds have turned one bid down for the 19-year-old, who has also been linked with Liverpool, West Brom and Wigan.
The Mirror report that Chelsea would pay about £1m for the player, who is out of contract at the end of the season.
Carson's exit will pay some vital bills, including a tax demand and allow Leeds a better chance of avoiding administration before the end of the season.
Carson will probably only be the No 3 keeper at Stamford Bridge behind Petr Cech and Carlo Cudicini, but his reported £1m-a-year wage will compensate for that.