The museum announced hope last year to leave its current building on London Wall and move into the empty Smithfield General Market buildings, with an aim of opening in the new location in 2021.
Six architectural teams' designs from an international competition have made it onto the shortlist for the new museum.
The site being considered by the museum includes Smithfield General Market, the Fish Market, the Red House and the Engine House.
The proposals will go on show at the museum from Friday and will be studied by a panel headed by broadcaster Evan Davies, who is on its board of governors. A decision is expected this summer.