The 24-year-old made a good impression on Saints boss Gordon Strachan during a trial at St Mary's last October.
A number of clubs, including Tottenham, have been watching Smith, but Southampton are hoping to beat them to the chase by agreeing a deal with Brentford.
The Bees are reluctant to lose their shot-stopper and are set to reject Southampton's bid as they rate him in the £800,000 bracket.
"I've had no communication with Gordon Strachan, but we have received an offer from Southampton which I am obliged to take to the board," said Bees boss Wally Downes.
"In my view it is nowhere near the estimation that we have put on the player and my recommendation would be that it is insufficient."
