The centre, in Monkmoor Road, could be re-located to the A&E department at the Royal Shrewsbury Hospital, where an urgent care centre will be created instead.
The proposals, which have been put forward by Shropshire Clinical Commissioning Group (SCCG) would see the centre move for a period of up to three years.
The group said evidence suggests patients who attend A&E could actually have their needs met by a GP. The centre would be open seven days a week.
Patients will now be consulted on the move before the matter is discussed at a health scrutiny committee and a report suggests members should support the move.
Dr Caron Morton, accountable officer for SCCG said: "We have started to gauge the opinion of patients who use this valuable service on whether they think it would be beneficial to make the move.
"As commissioners we see definite advantages for patients. In the most recent national report reviewing emergency and urgent care there is evidence that a significant proportion of patients who go to A&E could have had their needs met by seeing a GP or other healthcare professional.
"If the move goes ahead the urgent care centre would be located alongside A&E at the Royal Shrewsbury Hospital seven days per week from 8am to 8pm.
"It would mean people who have a minor injury or ailment who may go along to A&E would be able to be seen by the right health professional first time. It would also free up A&E staff to see those patients that need more specialist or emergency care.
"We believe this will improve care for patients and it reflects the national review."
The Monkmoor Road walk-in centre was initially set up in 2009, and enables patients to receive treatment from a GP without first making an appointment,
It is currently run by Malling Health and the move has been suggested as the five-year lease for the centre will run out later this year.
The centre is used by 34,000 walk in patients and just under 3,000 patients are officially registered.
If the move goes ahead, the walk-in centre will relocate in November this year, and will coincide with the Woman and Children's Services moving to Princess Royal Hospital in Telford.
A decision will be made on July 14.