The pair, who have clocked up a total of more than 70 years at the BBC, took voluntary redundancy and will leave next year.
Their departures are a result of a shake-up in Radio 4's presentation team, which is being reduced by two posts to 10.
Ms Green, who said she had thoroughly enjoyed her 34 years at the corporation, has famously been gripped by fits of the giggles while on air, perhaps most notably after hearing the oldest-known recording of a human voice during a news item.
She said: "I'll miss the buzz of live radio, but I'm really looking forward to getting involved in new projects."
Ms Cass, who first joined the BBC in 1972, said: "The BBC was my first job and I thought it would be interesting for a few months. Forty years later, it's time to do other things."
