Both players will require surgery on the problems and will miss the opening stages of the season as a result, returning home early from the club's pre-season training camp in Switzerland.
Unconfirmed reports on Twitter have suggested that the injuries were as a result of both players returning home drunk after curfew and subsequently trying to climb into their hotel room via a balcony, with Carroll allegedly falling and rupturing his ACL.
"We are obviously very sad to hear that Andy and Winston have suffered setbacks. This is a big blow for them and us, but we hope to see them back in action as soon as possible," manager Manuel Pellegrini told the club's official website.
"It does mean that we have lost two 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."
West Ham's new head of medical Richard Collinge added: "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, following the 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."
West Ham will begin the new campaign away to Liverpool on August 12.