The Hammers are keen on signing the 24 year old, who has made 110 appearances for Saints since joining on an initial loan from Man City in 2023.
Saints paid £20million to make his signing permanent when they won promotion from the Championship in May 2024 under boss Russell Martin.
He scored on his senior England debut in November 2024, and has remained a key player for Saints this season, with 24 appearances and four goals.
West Ham made an eight-figure offer to purchase the defender, but it falls below that fee and is less than Saints would accept, particularly in January.
The fresh offer also included payment terms deemed to be unacceptable by Saints, with an extended payment plan over a period of many years.
Several media reports elsewhere have reported that Harwood-Bellis is open to making the move to London Stadium, but Saints are reluctant to sell.
The move is considered unlikely to happen, unless there is a significant increase in the bid. Harwood-Bellis is under contract until at least June 2028.