A display at the existing gallery has attracted 3,400 visitors since it opened in October - a threefold increase on the same period last year.
The borough council wants to relocate the museum and art gallery to a new bigger building.
It says the show of local support will help with a Heritage Lottery fund bid.
Curator Sophie Cummings, said she was pleased with the number of visitors.
"Even if you're not the biggest fan of art, Turner is a name you recognise, and I had a feeling Turner in Swindon would get people excited.
"But we've been really, really pleased with the number of people who've come through our doors."
Earlier this year, Swindon Borough Council launched a project to relocate the town's museum and art gallery to a new building in the town centre.
A spokesman said it would allow more of the collections it owns to be put on display, and have better access and facilities.
A bid for Â£12m will be submitted to the Heritage Lottery Fund, which Swindon Borough Council has pledged to match with another Â£5m if successful.
The landscape painter JMW Turner is one of Britain's most celebrated artists.
The visiting exhibition, called Turner - Watercolours from the West, has been on display since 15 October.
Campaigners hoping to build a new art gallery in Swindon say a huge number of visitors to a Turner painting exhibition has helped the cause.