The Museum of London plans to move from the Barbican to Smithfield Market by 2021 as it has run out of space.
The three proposed sites, which are currently derelict, include Smithfields General Market, the market's basement and the red-brick Red House, built in the 1800s.
In total, the new museum would cover 27,000 sq ft instead of the current 17,000 sq ft.
The plan, which is in its early stages, aims to double annual visitor numbers.
Andrew Marcus, a spokesman for the museum, said they had more artefacts than they could display, meaning many were not given justice, such as the first piece of stone the word London was written on from 160 AD.
He said it is "tucked away in a corner".
The situation also meant many items were in storage - such as the remains of 20,000 people at the London Wall site and the world's largest archaeological archive at Hackney, he said.
The museum explores the history of London dating back to 4,500 BC.
Its collections include Roman archaeological discoveries, a medieval gallery, bricks burned in the Great Fire and an original, art deco lift shaft from Selfridges.
Museum of London director Sharon Ament said there would be "many issues to be resolved" before a new one opens.
'Desperately' needs space
She said she would now look into how and if it can relocate with the Greater London Authority and City of London Corporation.
She added: "We have the aim of creating an unrivalled experience and contributing to the regeneration of a vibrant and historic part of London. "
London Mayor Boris Johnson said the museum told the "incredible story of this great city".
He added: "The London 2012 cauldron, the head of Serapis found in the Temple of Mithras, the Cheapside Hoard and the original lift from Selfridges are just some of the objects held by this cultural gem, but it desperately needs more space."