The former Liverpool trainee, who still has 15 month to run on his contract at Gay Meadow, was handed a trial last week at English Second Division club Wrexham.
The 23-year-old has been on The Shrews' transfer list since October and, although he desires to remain at the Conference club, Dunbavin has decided to try his luck with the Scottish outfit.
"I never wanted to leave Shrewsbury in the first place but I've been told that I have to," said Dunbavin.
"At this stage of my career, I need to be playing regular first team football.
"I enjoyed it at Wrexham because it was a different atmosphere and environment and they've said I can go back there, if I want.
"I'm still contracted to Shrewsbury until the end of next season and unless anything comes up that I want to take, I'll still be there."
