The combined authority announced the proposal on Thursday, saying it would add to £3.25 million already committed to the project. The new sum remains subject to the authority's decision-making process.
Bath and North East Somerset Council plans to reopen the museum in 2030 in the former Old Post Office. The scheme is intended to create a permanent home for a collection of more than 100,000 objects.
The authority says the completed museum could attract about 250,000 visitors a year. Its published assessment estimates £111 million in social and economic value and about £25 million in visitor spending over the appraisal period.
The project also includes learning and community activity, with an ambition to involve 7,800 people. Supporters argue that the museum can strengthen Bath's cultural offer while contributing to jobs and city-centre footfall.
Godwin's announcement is a proposed allocation rather than confirmation that all the money has been awarded or spent. Delivery also depends on the wider project's funding, construction and reopening timetable.
If approved, the investment would give the West of England mayor a prominent role in a cultural regeneration project designed to benefit both Bath and the wider regional economy.
