Watts, who was diagnosed in June, is finishing up six weeks of radiation therapy at a Chelsea hospital near his home in west London, BBC reported.
"He is expecting to make a full recovery and start work with the rest of the band later in the year," a band spokesman told the network.
Watts, 63, quit smoking back in the 1960s, the BBC said.
He reportedly is doing well and walking to the hospital for his treatments.
The Rolling Stones are on a break from touring and recording.