The Jam, formed in 1972, had a string of top 40 singles in the UK between 1977 and 1982, including four number one songs - among them, Going Underground, Beat Surrender and Town Called Malice.
"I'm shocked and saddened by Rick's passing," Weller wrote on Instagram.
"To all the pubs and clubs we played at as kids, to eventually making a record. What a journey!"
"We went far beyond our dreams and what we made stands the test of time," he added.
The Jam's bass player, Bruce Foxton, said he was "devastated to hear the very sad news today".
"Rick was a good guy and a great drummer whose innovative drum patterns helped shape our songs."
The 69-year-old musician added: "I'm glad we had the chance to work together as much as we did.
"My thoughts are with Leslie and his family at this very difficult time."
Buckler and Foxton later played together in a successor band, From the Jam, while Weller continued his career with the Style Council and as a solo artist.
Buckler had recently been forced to cancel a spoken-word tour of UK venues because of health problems.
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