It is understood City have met the Bees’ valuation for the 24-year-old, which is believed to be around £9 million.
Gray was also a target for Hull, but they were unwilling to meet the asking price.
City manager Steve Cotterill turned his attention to Gray after Crystal Palace forward Dwight Gayle turned down a move to Ashton Gate.
The former Luton frontman hit his second goal of the season as Brentford beat City 4-2 on Saturday.
Cotterill told the BBC at his pre-match press conference on Tuesday: “It will be nice if it were to come off, but I’m afraid there isn’t much I can say on that at this moment in time.”
Gray scored 18 goals last term, spearheading the Bees’ surprise run to the Championship play-offs, where they were eventually beaten by Middlesbrough in the semi-finals.
Boss Marinus Dijkhuizen, who replaced Mark Warburton at Griffin Park this summer, wants to keep Gray at the club but the Bees will sell at the right price.
They have already lost highly-rated right-back Moses Odubajo, who joined Hull for £3.5m earlier this month.