An article by Mark Palmer in The Times claims Turner is close to a deal that would see him link up once more with former Glasgow head coach Dave Rennie.
Turner joined up with Warriors in 2017 and went on to make his Scotland debut in November of that year.
The article goes on to say that Fraser Brown is set to announce his retirement from rugby.
The Scotland hooker hasn't played this campaign after sustaining a knee injury while playing for the World XV against the Barbarians at the end of last season.