The 30-day consultation, which began today, could see 90 staff lose their jobs, although a spokesman for the company said that it would attempt to find positions elsewhere in the company.
If the closure goes ahead, a number of titles, including the Exeter Express & Echo, would be moved to other Harmsworth sites. It is thought that most would go to its Didcot site.
According to Unite national officer Steve Sibbald, the closure is no surprise. He said: "The days of most of the Harmsworth printing sites are very much numbered, the titles will end up with someone like Trinity or News International.
"It will keep happening this year and next year until the UK has levelled out the market for newspaper printing. Hopefully we will hit a plateau soon."
The closure is the latest in a long line for Harmsworth, which has seen the company's plant list more than halve.
In June last year, the closest Harmsworth plant to Plymouth at Bristol was closed, with 88 staff losing their jobs.
Grimsby and Leicester were also both closed in 2009. Hull was shut a few years previously.
The company's two flagship sites at Didcot and London are still in operation, with Derby and Stoke the only two satellite facilities remaining.