Lee, who currently plays for Sanfrecce Hiroshima in the J.League, flew into London on Tuesday ahead of his move to Southampton, who are top of the Championship, England's second tier.
The Sports Nippon reported the 26-year-old would join the side on England's south coast on a two-and-a-half-year contract with the annual salary £800,000, almost double what he earns with Hiroshima.
If Southampton get promoted to the Premier League this season, Lee's pay will shoot up to £1.2 million, the sports daily reported on its website.
"I want to play in the Premier League, the world's best," Lee was quoted as saying before leaving Japan.
Lee will sign after getting a visa and undergoing a medical, reports said.
Jiji Press said he was expected to train with Southampton during his initial 10-day stay with the team.
Born to Korean parents in Japan, Lee scored the winning goal as Japan lifted a record fourth AFC Asian Cup last year. Appearing as an extra-time substitute, he volleyed home the winner in the 109th minute for a 1-0 win against Australia.