Swiss firm Diener & Diener Architekten and Paris studios Lacaton & Vassal and Studio Milou also made the shortlist for the new museum building at West Smithfield, which will replace the existing facility at the Barbican Estate.
The new 25,000-square-metre building will allow the Museum of London to more than double its one million annual visitors, and to show some of the artefacts from its six-million-piece collection for the first time.
It will take over the sites of several disused Victorian buildings that surround the historic market area, including the former Smithfield General Market, the Fish Market, the Red House and the Engine House.
Announced in April, each of the six shortlisted entrants leads a team of local architects and consultants. No specific details of their proposals have been shared yet, but competition organiser Malcolm Reading Consultants has revealed two images of each design.
Stirling Prize winner Stanton Williams is working with architect Asif Khan and conservation specialist Julian Harrap on its proposal, which features a generous public space at its entrance and a grand staircase.
Meanwhile, Lacaton & Vassal's team, which includes local offices Pernilla Ohrstedt Studio and Allies and Morrison, has proposed a museum with a focus on London's hidden spaces, and Studio Milou suggests a mixture of indoor and outdoor areas.
"It is clear from their concepts for a new museum that the architectural teams have been equally captivated by Smithfield's vivid history and vibrant character," said museum director Sharon Ament.
The Museum of London has been based in its current Powell & Moya-designed building, which forms part of the Brutalist Barbican estate, since 1976. It was founded in 1965, following the merging of the Guildhall Museum and the London Museum.
Its collections include paintings and photographs of London, the world's largest archaeological archive, historical artefacts and documents, and fashion and textiles.
The winning concept for the new museum building will be selected by a panel of judges and announced later this summer.
"These six concepts, which suggest a fascinating range of options, will give the jury plenty to consider when deciding upon an architect to work with us to design the new museum," added Ament.
The winner will be expected to submit a planning application in 2017 and to deliver the new museum by 2021.