The 28-year-old played for the Championship club in the 3-2 friendly win over Rotherham.
However, Mowbray told BBC Tees that any deal depended on reserve keeper Jason Luetwiler moving on.
"If Luetwiler was to go out on loan then we'd need another goalkeeper to cover so that would be the only option," the Boro boss said.
"There's a few things going on in the background. I spoke to Luetwiler about going out on loan and testing himself as he went a whole season last year and didn't play in our team - he played one cup game, I think.
"He's [Turnbull] been in training for a couple of weeks. He's without a club and that's all we should read into it at the moment.
"It's a pretty decent option for cover, an experienced goalkeeper who's been at Chelsea."