The duo are much needed reinforcements, given the Hammers have lost four games on the bounce since the Premier League season began.
Boss Manuel Pellegrini may have to wait a few weeks before integrating either player into his first team squad.
The club's head of medical, Richard Collinge, spoke about the two players to the club's official website.
"Andy Carroll is progressing really nicely and he's going to start doing some work on the pitches this week," said Collinge.
"Winston Reid is in a knee brace and he's eight weeks down the line from his surgery and making good progress.
"He's going to be a little bit longer, but he's also making good progress."