The £4.9m grant will be put towards the £42m revamp project, with the new museum expected to be open in 2022.
Renfrewshire Council has already set aside £24.1m towards the scheme, which will be included in a bid for Paisley to be UK City of Culture 2021.
Plans include an extension to the current Victorian-era building housing a cafe and shop, improvements to all four museum buildings including the Coats Observatory, changes to physical access, and an internal redesign which will double the number of objects on public display.
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In addition, 138 jobs are expected to be supported during the construction phase. The £4.9m HLF grant will be confirmed once a successful stage two application is lodged with HLF.
Renfrewshire Council leader Iain Nicolson said: "This is a wonderful piece of news from HLF and couldn't have come at a better time, with our bid to be UK City of Culture 2021 being lodged this week.
"The museum plans are central to our wider vision to transform the town's future by harnessing the power of Paisley's internationally-significant heritage and cultural story.
The council's bid to be UK City of Culture 2021 will be submitted to the UK Government on Friday, 30 September.
(LM/MH)
