Ajax said they had agreed on transfer fee of 43 million euros, which "can increase through variables".
"I've been dreaming of this moment and I'm very happy to be here. But it doesn't end here, I want to move on," Kudus said in a statement.
He arrives at the London Stadium having made 87 appearances, with 27 goals and 12 assists, during his three-year spell at Ajax, where he won the Eredivisie twice (2021 and 2022) and became Dutch Cup champion in 2021.
The Ghana international has also played 24 times for his country, scoring four goals and appearing in all three group stage matches at last year's Qatar World Cup.
Kudus will join David Moyes' side as the third major signing of the summer following former Ajax teammate and Mexico international Edson Alvarez (25), along with former Southampton midfielder James Ward-Prowse (28).
