The Hammers are left Vladimir Coufal as their sole right-back after seeing Ben Johnson join Ipswich Town on a free transfer earlier this summer.
West Ham initially identified Bayern Munich's Noussair Mazraoui as their top full-back target before Man United entered the race for the Morocco international.
The Hammers soon turned their focus to Man United's Wan-Bissaka, who has entered the final 12 months of his contract at Old Trafford.
It was recently reported that the two clubs had made progress in negotiations over a permanent transfer for the former Crystal Palace man.
Man United agree Wan-Bissaka settlement
According to the Daily Mail, Wan-Bissaka is set to undergo a medical on Monday after agreeing a settlement with Man United.
The report claims that the defender was set to have a medical on Sunday, but it was postponed as discussions took place over the exit settlement.
Having resolved the issue, Wan-Bissaka is now set to complete a £15m move to West Ham which will bring an end to his five-year stay at Old Trafford.
The full-back is set to leave the Red Devils after making 190 competitive appearances, helping the club win the FA Cup and the EFL Cup.
Wan-Bissaka's departure to pave way for Mazraoui arrival
Man United have been waiting to offload Wan-Bissaka before they proceed with the final steps of Mazraoui's proposed move.
With Wan-Bissaka on the verge of joining West Ham, Man United are now set to sign the Moroccan for an initial €15m (£12.9m) plus €5m (£4.3m) in add-ons.
Mazraoui, who has spent the last two years at Bayern, will put pen to paper on a five-year contract with an option to extend by a further 12 months.
Man United are expected to unveil Mazraoui and his Bayern teammate Matthijs de Ligt as their latest summer additions on Monday.
Should everything go according to plan, the pair could feature in the matchday squad for Friday's Premier League opener against Fulham.