The Spaniard has flopped since covering for the injured Lukasz Fabianski, and the Hammers see Begovic as a quality replacement and experienced hands capable of providing back-up and competition for their first-choice.
The 32-year-old is currently on loan at Azerbaijan with Qarabag, but his deal expires on December 31st, and the likes of Derby County, Birmingham City and PSV Eindhoven are all monitoring him.
The Bosnia international is keen to join a new club temporarily or permanently, and West Ham could be landing a top goalie in him.
Begovic is a Premier League title winner from his days at Chelsea where he made 33 appearances across all competitions in two seasons, and he currently has more than two years left on his £60,000-per-week contract with the Cherries.
West Ham would be expected to pay a large part of his wages should they secure his services on loan, and it will be interesting to see if they indeed go after him in January.
David Martin has been Manuel Pellegrini’s first-choice in goal for the last two games, but Fabianski is expected to return to action next month, and West Ham could look to bring in a new second-choice with Roberto set to leave.