The firm will refurbish the council's Crookhill offices on Chickerell Road in addition to carrying out interior alterations to a former children's nursery on Commercial Road.
The renovations are part of a wider plan to downsize the council's main office at North Quay to save £2 million in essential building repair work and annual running costs.
Works on Chickerell Road involve creating additional office space for around 50 council workers.
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On Commercial Road, refurbishment work will create a mayoral parlour, several meeting rooms and a public reception room.
The new office space will be used by council staff currently based at the council's main North Quay office.
The site has outline planning permission for a creative scheme based on a view of the original harbourside consisting of a mixed use development of 72 homes and 216 sq m of commercial space.
Cllr Jeff Cant said: "After a rigorous tendering process, we are looking forward to Morgan Sindall delivering our new town centre, cheaper-to-run offices, on time and to the high standards for which the firm is known.
"We have consulted widely to try to create an office that meets our residents' needs. The new offices will be mainly appointment based but will still cater for residents needing to come in on an ad-hoc basis."
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