The charity that runs institutions named after Edward Colston says it is "right" his statue has been removed.
The slave trader's statue was pulled down and thrown into Bristol's harbour during a Black Lives Matter protest last weekend.
The Society of Merchant Venturers said it was time to acknowledge Bristol's "dark past".
Bristol's Colston's Girls' School also says it will hold a consultation on changing its name.
The society, which dates from the 13th Century guild that funded John Cabot's voyage to North America, helps run schools and charities in Bristol.
In a statement, it said: "The statue of Edward Colston was removed from Bristol's city centre last weekend and the fact that it has gone is right for Bristol.
"To build a city where racism and inequality no longer exist, we must start by acknowledging Bristol's dark past and removing statues, portraits and names that memorialise a man who benefited from trading in human lives."
In a letter to parents, Colston's Girls' School, which is part of the Merchant Venturers Education Trust, says a six-week consultation on its name change will begin in the autumn.
The school said: "Following the protests in Bristol last Sunday, many of our students, both past and present, have contacted the school to share their feelings, including about the name Colston's Girls' School, which some students wish to keep and others wish to change.
"We are very open with our students about where the wealth of the school's benefactor came from and we recognise that the name Colston does not always sit comfortably."
Since the Colston statue was pulled down by protesters, the nearby Colston Tower has also removed his name.
Colston Hall, the city's largest concert venue, said in 2017 that it would be changing its name when it reopens after a £49m refurbishment.