While neither team has yet to confirm the trade, Puig tweeted his thanks to the Reds, their fans and the city of Cincinnati.
The trade is part of a three-team deal that also includes the Padres.
Bauer heads for Cincinnati with a 9-8 record through 24 starts this season. The 28-year-old's playing in his second consecutive arbitration year, earning $13 million in 2019 and will be under team control next season.
Reds reliever Amir Garrett, being lifted after allowing a ninth-inning home run, charged at the Pirates dugout Tuesday at Great American Ballpark.
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Reyes, a rising star for the Padres, broke through past year thanks to injuries to Will Myers and Hunter Renfroe, making four trips between San Diego and Triple-A El Paso but quickly proving his value at the plate down the stretch. He initially chucked a ball at the netting behind home plate, and then when manager Terry Francona came out to pull him from the game he launched another out of the stadium. Puig will be a free agent after the season, so he is a rental for an Indians offense that could use some extra power. Allen, 22, made his major league debut on June 18.
Allen, 22, is 2-3 with a 6.75 ERA in four starts (eight appearances) for San Diego.
Part of the math being done by the Indians' front office in trying to make a deal involved the near imminent return of two injured starters: Corey Kluber and Carlos Carrasco. "And then, if there is a time where everybody says that he can pitch, he won't have been that far off, because he's been throwing". "We already had a legit pitching staff, it just got more legit".
After a breakout 2018 season in which he made his first All-Star team, Bauer hasn't been as consistent as Cleveland has needed while waiting for two-time Cy Young Award victor Corey Kluber (broken arm) and Carlos Carrasco (leukemia) to return to their rotation.