Hammers deal-maker Tim Steidten flew out to Amsterdam on Thursday for talks with Ajax officials as he looks to take Kudus to the London Stadium.
Ajax are open to letting Kudus depart for the right price and the player, who has already agreed personal terms with West Ham, is ready to move.
Now the deal is a step closer as West Ham are close to a full agreement with Ajax, according to French outlet Foot Mercato.
The deal to take Kudus to West Ham would come in at around £35m.
West Ham will aim to put the finishing touches to an agreement with Ajax and then quickly complete the remaining steps needed.
Once the Kudus deal is sealed then West Ham will switch their focus to snapping up Rennes winger Jeremy Doku.
Doku is also wanted by a number of other Premier League sides, but the Hammers will hope they can steal a march on them and win the race.
