The 23-year-old Gakpo scored three goals for the Netherlands in the World Cup in Qatar, where the team were beaten by eventual champions Argentina in the quarter-finals.
The clubs did not disclose a transfer fee but British media reported it was around 40 to 50 million euros ($43 to $53 million).
PSV general manager Marcel Brands said the transfer fee would be a record for the club.
The 23-year-old attacker will immediately leave for England where he will undergo the necessary formalities to complete the transfer.
“Both clubs are not making any announcements about the transfer fee. ‘But this is a record transfer for PSV,’ said general manager Marcel Brands.
The PSV management completed the transfer on Boxing Day and gave Gakpo permission to travel to England. Gakpo will cost an initial £37million and the forward could complete a medical in the next 48 hours.
Jurgen Klopp hinted at January business when recalling the impact Luis Diaz made after signing in the last winter window.
He said: “We have two long-term injuries up front. Last January, Luis Díaz came in which was very helpful for that part of the season. Doing something similar this time? If it was a positive move, why not?”
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