Both players suffered injury problems last season, with Carroll playing a limited role under both Slaven Bilic and David Moyes while Reid played just three Premier League games in 2018.
New manager Manuel Pellegrini has aired his disappointment on losing both players, and said there may have to be further spending to replace two senior figures.
Both will return to London from their Bad Ragaz training resort and will need to undergo surgery.
“We are obviously very sad to hear that Andy and Winston have suffered setbacks,” Pellegrini told the club website. “This is a big blow for them and us, but we hope to see them back in action as soon as possible.
“It does mean that we have lost two senior players in key areas, though, and we’ll continue to work hard with the owners to make further additions to strengthen the playing squad, beyond the five new signings already confirmed.”
Newly appointed head of medical, Richard Collinge, said: “Andy played and scored in Sunday’s match but is still reporting some discomfort in his ankle and further scans have revealed that he requires some minor surgery to provide more strength and stability to the area.
“Winston has been suffering some swelling around the knee injury that he sustained at Swansea last season and he too is in need of explorative surgery to clean up the area. We have decided that this is the best course of action to ensure that both players have a chance of returning to full fitness as soon as possible.”