Gakpo, 23, had an impressive World Cup in Qatar, scoring three goals as he helped the Netherlands reach the quarter-finals.
The fee for the winger is expected to be between 40m-50m euros (£35.4m-£44.3m).
PSV say Gakpo will travel to England to complete the deal, adding that it would be a record transfer for them.
Liverpool have made a move with forwards Luis Diaz and Diogo Jota facing lengthy spells on the sidelines.
Defender Virgil van Dijk, a club and international team-mate, spoke glowingly about the winger during the World Cup.
"Hopefully Cody can go to the moon and back," said Van Dijk during the tournament in Qatar.