A paper being presented to the council's finance, resources and customer services policy board on 31 March will seek to authorise awarding the £23m contract to Kier.
If the paper is approved negotiations will move to the final stage with a view to formally appointing the contractor in the coming weeks. This would result in work beginning on site in early summer with construction likely to be completed by late 2022, reopening to the public in 2023.
A programme of community benefits was requested as part of the procurement process. If approved as the main contractor, Kier Construction has agreed to offer a number of local employment opportunities, work experience placements, business advice and support to local SMEs, social enterprises and voluntary organisations. There is also a commitment to ensure that supply chain opportunities are prioritised for Renfrewshire-based businesses.
The £42m redevelopment of the museum is the signature project in Paisley's regeneration, which also includes refurbishment of the town's A-listed town hall and a new learning and cultural hub housing library services on the high street.
Paisley Museum Reimagined project director Kirsty Devine said: "If the proposed contractor is approved by councillors next week it will enable us to move ahead with plans to realise our ambitious vision for Paisley Museum.
"The redevelopment of the museum provides the perfect opportunity to showcase our internationally significant collections and we're doing this in collaboration with Renfrewshire communities as well as with partners across the world. The redeveloped museum will be an amazing resource in the heart of Paisley and people should feel rightly proud of that heritage."
The revamped museum is expected to attract around 128,000 visitors a year when it reopens - creating new footfall and trade for Paisley town centre and supporting new and existing businesses in the surrounding area.
Chair of Paisley First, Colette Cardosi, said: "We welcome what would be a major step forward in the museum redevelopment, a project we know will enhance the reputation of Paisley town centre as a must-visit destination.
"Like everyone who is passionate about the town, it will be great to see real progress at the site and we are looking forward to watching the transformation as the project develops."
The redesign of the museum campus, being led by an international team including architects AL_A and exhibition designer Opera Amsterdam, will allow the number of objects on display to be increased by 150%.