Multiple US media outlets reported that Diamondbacks ace Greinke, a six-time All-Star and former Cy Young Award winner, was heading to Houston in exchange for a raft of top prospects.
Greinke, 35, boasts a 2.90 ERA this season with 10 wins and four losses from 23 starts.
The Astros are currently top of the American League West division with 69 wins against 39 losses.
There was no immediate confirmation of the deal from the Astros or Diamondbacks.
If confirmed, Greinke's move looked set to be the biggest of a low-key trading window.
Despite frenzied speculation prior to the trade deadline, neither the New York Yankees nor the Los Angeles Dodgers strengthened their pitching with an elite-level signing.
The San Francisco Giants, meanwhile, who have surged into wild card contention in the National League West, held on to three-time World Series champion pitcher Madison Bumgarner and highly rated reliever Will Smith.
Bauer, who angered Indians manager Terry Francona on Sunday after hurling a baseball over centerfield in frustration, joins the Reds after six years in Cleveland.
Under the three-team move, the Reds' Cuban outfielder Puig and minor league pitcher Scott Moss will move to Cleveland.
The San Diego Padres are sending outfielder Franmil Reyes, left-handed pitcher Logan Allen and outfielder Victor Nova to the Indians.
Reds outfielder Taylor Trammell heads to San Diego as part of the deal.
Bauer and Puig's moves come after recent controversy involving both players.
Bauer irked Francona on Sunday after hurling a pitch into center field in disgust.
Puig's last act for the Reds, meanwhile, saw him pile into a mass brawl on Tuesday in a clash with the Pittsburgh Pirates.
"We appreciate Trevor's contributions to the organization in his time with us, and while it's never easy to part with a player of his caliber, we feel we've traded from an area of depth to help bolster our Major League club for this year and the foreseeable future," Indians president of baseball operations Chris Antonetti said in a statement.
Bauer, an All-Star in 2018, was part of the Cleveland team which reached the World Series in 2016, losing in seven games to the Chicago Cubs. Bauer lost both of his starts on the mound in that series.