ESPN reported the Red Sox will send outfielder Josh Reddick, third baseman Miles Head and pitcher Raul Alcantara to Oakland.
Bailey would replace Jonathan Papelbon in the Boston bullpen after Papelbon bolted the Red Sox to join the Philadelphia Phillies.
Bailey, who garnered 2009 American League Rookie of the Year honors, compiled 75 saves and a 2.07 ERA in 157 appearances over three seasons with Oakland.
Sweeney confirmed the trade to the San Jose (Calif.) Mercury-News, saying he was excited but surprised the trade after Oakland had tendered him a contract this month.
The key for the A's in the trade is Alcantara, The Boston Globe reported.
The 19-year-old prospect has a 2.72 career ERA over two minor-league seasons and is hard-thrower, the newspaper said.