Whitechapel Bell Foundry, which opened in 1570, is in the Guinness Book of World Records as Britain's oldest manufacturing company.
The company's owner Alan Hughes said he hoped to have found a buyer by the time he retires in May.
The Grade II-listed building in Whitechapel, where it has been based since 1738, has already been sold.
The foundry has made numerous famous bells including the Liberty Bell in Philadelphia which summoned the city's residents to hear the first public reading of the Declaration of Independence in 1776 and became a symbol for campaigners wanting to abolish slavery.
The foundry also cast the bells for Westminster Abbey, St Paul's Cathedral and numerous other cathedrals across the country.
Mr Hughes' grandfather bought the business in 1904.
"We have made this decision with a heavy heart, but in response to the changing realities of running a business of this kind," he told local community website Spitalfields Life.
"The Bell Foundry in Whitechapel has changed hands many times, but it has always been a family business.
"My own family has owned the foundry since 1904, but other families have run the firm throughout its history.
"The business has been at its present site for more than 250 years so it is probably about time it moved once again.
"We hope that this move will provide an opportunity for the business to move forward in a new direction."
Mr Hughes said the firm could no longer afford the upkeep of its Whitechapel building and recently had to spend Â£20,000 to repair a leak in the roof.
A foundry which opened in Queen Elizabeth I's era and made Big Ben is up for sale.