Panter Hudspith and Keith Williams have seen off competition from Stirling-shortlisted Caruso St John, as well as Tony Fretton and Jamie Fobert.
Also on the list for the scheme, a mixed-use development of the museum, housing and offices, was van Heyningen and Haward.
It is understood that the jury will now visit buildings by the final two, such as Keith Williams' Unicorn Theatre in south-east London.
The new museum will house 'significant geological, archaeological and historical collections' and will tell the story of the town's Roman origins, its medieval history and its Georgian townscape.
An honorarium of £1,500 was awarded to all six shortlisted firms.