West Ham already had a bid for Ritchie, and teammate Callum Wilson, turned down earlier this year.
Matt Ritchie celebrates scoring the first goal for Bournemouth
The Hammers are keen to add to their squad this summer, and have already signed Havard Nordveit and Sofiane Feghouli on free transfers.
West Ham manager Slaven Bilic
They are unlikely to be the only arrivals at the club as they seek to build on an impressive first season under Slaven Bilic.
They had already made a move for the Bournemouth winger earlier this year, offering £20 million for him and striker Callum Wilson.
The Cherries swiftly rejected that dual bid, and are keen to keep onto two of their key players next season. West Ham are determined to test their resolve though and are now poised to make another offer for Ritchie this summer.
The Telegraph claims they are prepared to pay an initial £8 million for his services, with that rising to £10million in instalments, likely to be based on performances.
Bournemouth's Matt Ritchie has a shot at goal
According to the report, West Ham contacted Bournemouth on Wednesday in an attempt to reopen talks over the player.
They were reportedly told not to make another bid, but the Hammers are confident a deal can be done and a second offer is likely to come in the next few days.
West Ham's manager Slaven Bilic looks dejected