Harry Maguire has been booed by sections of Manchester United fans during pre-season
Each deal is thought to be worth in the region of £30m.
Maguire, 30, struggled for first-team opportunities at Old Trafford last season and has been stripped of the captaincy this season.
He is understood to be keen on the move with personal terms yet to be agreed but not expected to be a problem.
Maguire joined United in August 2019 for a world record £80m for a defender.
West Ham have been trying to sign Ward-Prowse for much of the summer but Southampton, who were relegated to the Championship last season, had rejected their advances until this point.
However, it has appeared inevitable that Southampton would lose their captain and that now seems set to happen.
West Ham are looking to bolster their squad after selling England midfielder Declan Rice to Arsenal for £105m and Italy striker Gianluca Scamacca to Atalanta for £21.5m.
Hammers boss David Moyes is closing on signing Ajax's Edson Alvarez as a replacement for Rice and has also been linked with a move for Manchester United's Scott McTominay.
Maguire made 30 appearances for United last season, but only made eight league starts.
He had potentially slipped down to fifth choice central defender at Old Trafford, with Lisandro Martinez, Rafael Varane, Victor Lindelof and Luke Shaw ahead of him.
After stripping him of the captaincy Erik ten Hag said Maguire was an "important player" and could "fight himself into the team".
But the defender was booed by United fans and shouted at by new goalkeeper Andre Onana during a pre-season friendly against Borussia Dortmund.
Despite his struggles at Manchester United, Maguire has been a regular for England under Gareth Southgate, starting all five games at the 2022 World Cup.
Ward-Prowse started Southampton's Championship opener against Sheffield Wednesday on Friday but did not play in their Carabao Cup first-round defeat against Gillingham on Tuesday.
Rice's departure leaves West Ham without an obvious captain and Moyes has opted not to name a replacement so far, but Maguire and Ward-Prowse provide two options.
Current first choice central defenders Nayef Aguerd and Kurt Zouma have both had injury problems in recent months.
Premier League champions Manchester City are interested in Hammers midfielder Lucas Paqueta with an initial approach made but no bid yet.
-- Courtesy of BBC Sport