The BBC has apologised to Sachs, who played waiter Manuel in classic comedy 'Fawlty Towers', for the "unacceptable and offensive" content of the calls, which have generated more than 546 complaints.
A BBC Radio 2 spokeswoman said: "We have received a letter of complaint from Mr Sachs' agent and would like to sincerely apologise to Mr Sachs for the offence caused.
"We recognise that some of the content broadcast was unacceptable and offensive. We are reviewing how this came about and are responding to Mr Sachs personally. We apologise to listeners for any offence caused."
Sachs, who starred alongside John Cleese in the BBC classic comedy 'Fawlty Towers', was expected to be a guest on Brand's Saturday night BBC Radio 2 programme on October 18, but had to drop out due to "unforeseen circumstances".
When Brand found out that Sachs had pulled out he and Jonathan Ross, who was appearing as a guest on the show, left a series of explicit messages on the actor's answer phone about his 23-year old granddaughter Georgina Baillie who auditioned to be a Sun Page Three girl in 2005. Baillie, who lists "pole dancing and luvvvv" as her sporting interests on her Bebo page, has denied the claims.
Brand told his listeners: "In a minute we are going to be talking to Andrew Sachs -- the elephant in the room is, what Andrew doesn't know is, I've slept with his granddaughter."
In one message Ross shouted out: "He fucked your granddaughter." In another Brand sang: "I said some things I didn't of oughta, like I had sex with your granddaughter".
Brand delivered a light hearted apology on his radio show on Saturday night stating: "I'd like to take this opportunity to issue a personal Russell Brand apology to Andrew Sachs, the great comic actor who played Manuel, for a message that Jonathan and I left on his answerphone, but it was quite funny."
Sachs' agent Meg Poole said that the actor was "very upset" by the calls and that she had written a letter to Radio 2 controller Lesley Douglas asking for an apology.
Earlier this year Brand apologised for making a hoax call to the police, during a stage show in Northampton, claiming he had spotted a man who was responsible for a series of assaults.
In September, Brand hosted the MTV Video Music Awards in Los Angeles last month and has been asked to return next year after he sent ratings up by 20%. In his controversial performance Brand labelled US President George W Bush a "retarded cowboy".
Brand has been a regular presenter on Radio Two for almost two years.