However, Reds president of baseball operations Walt Jocketty said the phone hasn't been ringing from clubs reaching out about any of his players.
"Not yet," Jocketty said on Tuesday.
As the Aug. 1 non-waiver Trade Deadline gets closer, it would seem likely that players like Bruce and Cozart will be attractive to contenders in need of help as they push to get to the playoffs.
"Maybe. It's really hard to say," Jocketty said. "I don't think anything will happen much until after the Draft. That's usually when it starts to happen, a couple of weeks after the Draft."
This year's Draft is June 9-11, but trades this early are certainly possible. On Tuesday, the Blue Jays acquired reliever Jason Grilli from another rebuilding team in the Braves for a Minor League pitcher.
• In his second rehab assignment start for Triple-A Louisville on Monday, Reds pitcher Anthony DeSclafani had a nice showing as he gave up one run and one hit over four innings with no walks and seven strikeouts. Of his 62 pitches, 45 were strikes.
"A changeup for a homer to a left-handed hitter was the only blemish," Reds manager Bryan Price said. "He was really sharp, 19 of 21 sliders for strikes. Curveball, changeup for strikes, maintained velocity, threw 60-plus pitches."
DeSclafani is expected to get at least 80 pitches for his third, and possibly final, rehab start for Louisville on Saturday. If that goes well, he could be activated to start vs. the Cardinals next week.
• When Tony Cingrani failed to close out the ninth inning Monday and gave up three runs to the Rockies while recording two outs, it fell on JC Ramirez to get the final out for the Reds' 11-8 win. Ramirez got Dustin Garneau to fly out to right field for the third out that gave him the first save of his big league career.
Ramirez held on the ball from that out for a keepsake.
"It wasn't a lot of pressure for me. It wasn't like, 'It's the big leagues.' All you have to do is throw strikes. That's it," Ramirez said. "Now I've gone [and] got it out of the way. Now I can focus on pitching and keeping the game simple."
Mark Sheldon has covered the Reds for MLB.com since 2006, and previously covered the Twins from 2001-05. Read his blog, Mark My Word, follow him on Twitter @m_sheldon and Facebook and listen to his podcast. This story was not subject to the approval of Major League Baseball or its clubs.