The club has not announced the deal, but the outfielder's contract, according to the source, is worth $750,000 with plate-appearance incentives that can bring it to $1 million.
Gross hit .227 with six home runs and 36 RBIs in 282 at-bats while starting in 67 games in right field last year for Tampa Bay. He is a career .239 hitter with a .337 on-base percentage and .399 slugging percentage.
The 30-year-old Gross, who will enter his sixth big league season with his fourth team, presumably represents the A's top option for a fourth outfielder and gives Oakland a left-handed option off the bench. He has experience at all three outfield positions.
The signing of Gross gives the A's a 41-man roster, meaning another transaction will have to be made when the deal is officially announced by the club to make room for the newest outfielder.
Jane Lee is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.