Bruce, a two-time All-Star, hadn't previously been mentioned as a candidate for a trade, as Cincinnati is believed to be more interested in trading a starting pitcher.
According to the report, the Reds' trade efforts are still focused on moving a starting pitcher, but the team may decide to trade Bruce as well in order to increase payroll flexibility and help rebuild the team. The Padres are reportedly among the teams that have discussed Bruce with the Reds.
Fox Sports reports that Cincinnati management isn't sure if it should trade Bruce or hold on to him and hope for improved team health to result in a rebound season in 2015.
Bruce played in 137 games but was clearly affected by in-season knee surgery, hitting just .217 with a .654 OPS and 18 home runs. All three figures were the worst of Bruce's seven-year career.
The Reds signed Bailey to a long-term contract last offseason but the other four pitchers in the starting rotation -- Johnny Cueto, Alfredo Simon, Mat Latos and Mike Leake -- aren't signed past next season, nor is closer Aroldis Chapman. Cincinnati picked up the 2015 option for Cueto late last month.
The team is said to be looking to acquire a left fielder after declining Ryan Ludwick's option at the same time it picked up Cueto's.
Bruce, 27, was selected by the Reds in the first round of the 2005 MLB draft. In his career, Bruce has hit .251 with a .790 OPS, 182 home runs and 551 RBI.
The Reds went 76-86 last season and missed the playoffs, recording their fewest wins since 2008.
- Ben Estes