The centre in Sidwell Street s the subject of a review which also includes out-of hours services and minor injuries units.
Campaigners are preparing to present a petition to the local Clinical Commissioning Group next week.
There's a proposal for Exeter to have one or more urgent care centres providing these services, and various organisations are bidding for the contract.
Council fights unreleased Ofsted report
Authority still rated inadequate
£550,000 Christmas windfall for Brixham pool
'It's time to celebrate!'
Girl missing from Newton Abbot
Christeen Madge could be trying to go abroad
Exeter to bid for unitary status in major council shake-up
City's 'powerhouse' role key to county success
Bomb squad investigates suspicious car in Cranbrook
Homes not yet evacuated
