Many people believe the street is named after James Penny, a famed local who made his fortune in the slave trade. But others say that's not the case at all, reports Sky News.
Liverpool Mayor Steve Rotherham said, "If it is as a direct consequence of that road being called Penny Lane because of James Penny, then that needs to be investigated...Something needs to happen and I would say that sign and that road may well be in danger of being renamed. But, of course, there is no evidence that is the fact."
Mr Rotherham, who described himself as a "massive" Beatles fan who has "done a bit of reading on this", suggested that Penny Lane was instead associated with a toll that was once paid in that area - in pennies - to cross the road.
But he added, as a "58-year-old white middle-aged man", he can "see things through my eyes and I can't see through the eyes of somebody who every single day has faced prejudice, has faced racism."
"It's for other people to decide whether they think it's appropriate that road sign is taken down, if indeed there is any link to either slavery or other incidences," the mayor added.
Liverpool's International Slavery Museum has said evidence linking Penny Lane to James Penny is "not conclusive."
The museum is "actively carrying out research on this particular question", a spokeswoman added.
Until the abolishment of the British slave trade in 1807, ships from Liverpool carried around 1.5m enslaved Africans on approximately 5,000 voyages, the vast majority going to the Caribbean.