Perez - who earns £80,000-a-week, according to Spotrac - has not enjoyed the best spell since moving to the London Stadium in the summer. He has struggled for consistent game-time, and has scored just four goals in 13 appearances for Manuel Pellegrini's side.
The Daily Star (09/01; page 43) reports that the Hammers are keen to move Perez on this month, and the striker is attracting loan interest from Real Betis. And it appears that they will look towards Anfield if Perez does end up leaving.
The Star suggests that Origi is being targeted as a potential replacement for Perez.
The 23-year-old has seen his stock rise in recent weeks thanks to a dramatic winning goal in the Merseyside derby in December and a superb finish during the Reds' FA Cup defeat to Wolves on Monday.
There will be some Irons fans who are sceptical about the logic behind a move for the Belgian. Few expected him to get another chance in the Liverpool first-team after an underwhelming few years on their books.
His recent performances may have provided Pellegrini with encouragement, but Origi would still have a lot to prove if he did end up moving to West Ham to replace Perez this month.