The 34-year-old, who was released by the Bluebirds earlier this summer, has been been training with Mark Warburton's side for the past few days.
"I've known Tommy since we were both at Watford and I know just how good he is," Sky Sports News quotes the Bees manager as saying. "We are having a look at him, and he is having a look at us."
"He is a model pro who looks after himself and is still unbelievably fit. He wouldn't still be with us if we were going to rule him out due to his age.
"We have some very talented young players at this club, but especially moving up a level, you need to make sure you have some wise old heads too."
Smith has also had spells at Queens Park Rangers, Sunderland, Derby County and Watford.