The headstones, both in Liverpool’s Allerton Cemetery, were defaced with graffiti overnight, a spokesman for the city council said on Tuesday.
Sir Ken's headstone has had to be removed to be further cleaned, while Black’s headstone was repaired on site, according to the Liverpool Echo.
The graffiti was said to be “offensive” on one community Facebook group, but the exact nature of the vandalism has not been confirmed.
The council spokesman said: “We were appalled to discover this morning that the graves of Cilla Black and Sir Ken Dodd had been vandalised overnight.
“Staff are carrying out repairs.
“We would urge anyone who has any information about the person or people responsible to contact the police.”
Liverpool-born Black died in her Spanish holiday home in 2015, aged 72.
Her body was brought back to the UK and a funeral service was held at St Mary’s Church in Woolton, attended by stars including Sir Tom Jones and Andrew Lloyd Webber.
The service was followed by a private burial at the cemetery, where she was laid to rest alongside her mother Priscilla and father John.
Comedian Sir Ken died aged 90 in 2018 at his home in Liverpool’s Knotty Ash, where he had lived all his life.
Thousands gathered for his funeral at Liverpool Cathedral, before a private interment was held.
The much-loved star, famous for his epic stand-up shows, died just two days after marrying his partner of 40 years Anne Jones.
His publicist Robert Holmes said at the time: "To my mind, he was one of the last music hall greats. There is no-one else that comes close.
"He passed away in the home that he was born in over 90 years ago. He's never lived anywhere else. It's absolutely amazing.
"With Ken gone, the lights have been turned out in the world of variety.
"He was a comedy legend and a genius."