Per ESPN's Jeff Passan, Reddick told reporters Friday that one person "wished cancer upon his children" and other Astros players have also been threatened.
Per Chandler Rome of the Houston Chronicle, a threat directed toward Reddick's child was posted as a comment on a video of one of his five-month-old twins.
After meeting with Astros players Friday, MLB Players Association Executive Director Tony Clark told reporters "there's a heightened sense of concern to make sure that themselves and their families are protected."
Reddick's comments come a day after Oakland Athletics pitcher Mike Fiers, who blew the whistle on Houston's cheating scheme, said he received death threats.
"Whatever, I don't care. I've dealt with a lot of death threats before. It's just another thing on my plate," Fiers, who played with the Astros from 2015-17, told Susan Slusser of The San Francisco Chronicle.
Reddick signed with the Astros after the 2016 season. He won a World Series in his first year with the club when they defeated the Los Angeles Dodgers in seven games.
The validity of that championship has been called into question in the wake of MLB's investigation that determined Houston's cheating scheme was in place throughout 2017 and part of the 2018 season.