The 24-year-old has attracted interest from a host of clubs, including Premiership outfits Portsmouth and Tottenham Hotspur.
However, it is The Saints who look set to land the Griffin Park custodian this week, as the St Mary's club look to boost their goalkeeping department, with Paul Jones likely to leave the South Coast outfit in the future.
Smith was left on the bench on Saturday as the South West London outfit went down to a 1-0 home defeat to Wrexham.
Following the match, Downes revealed the club's decision to accept Southampton's bid for Smith was purely for financial reasons.
"I think he's going for a medical on Monday," Downes told Brentford's official website.
"As far as I'm aware, the transfer to Southampton will go through from there.
"The top and bottom of it is we needed the money to survive as a club, and he had to be sold.
"Although it was a slim chance, because of the transfer window, time was running out to do the deal, and if we had not done the deal, who knows were we would have been financially."
