Complaints have flooded into the corporation demanding action over the string of offensive messages left by Brand and Ross on 78-year-old Sachs' answerphone, which was broadcast to 2m people on Brand's BBC Radio 2 show on October 18.
In the pre-recorded messages, Brand claimed he "fucked" Sachs' 23-year-old granddaughter Georgina, singing: "I said some things I didn't of oughta, like I had sex with your granddaughter".
The BBC initially defended Ross and Brand, but later released a statement calling the stunt "unacceptable and offensive". The two comedians have also apologised to Sachs.
However, media watchdogs and members of parliament have joined the chorus of complaints, some calling for an investigation.
Ofcom said it had received a numbers of calls from members of the public who were offended by the broadcast and will assess the complaints to see if further investigation is warranted.
Mediawatch-UK director John Beyer said he also received complaints and has since called for the BBC to remove Brand and Ross from programming while the incident is investigated.
Conservative MP John Whittingdale, who also chairs a media watchdog committee, said: "This is a serious failing on the part of the BBC. They claim to have stringent editorial controls and yet this got past all of them.
"This is the latest in a succession of appalling incidents from both comics. They seem to repeatedly cause offence and get away with it."
Another Conservative MP, David Davies, called for the BBC to sack Brand and Ross: "In no other organisation would you be able to get away with behaving like this."
It emerged yesterday that the radio programme was green-lit by 25-year-old BBC producer Nic Philips without the okay from Sachs', who Philips called to see if the programme could go to air.
The BBC has not revealed whether any further action will be taken over the complaints.