Sky sources understand a deal for the 22-year-old was finalised last month, and Spurs have now confirmed his arrival on a five-year contract.
An Austria international, centre-half Wimmer helped Cologne win promotion to the Bundesliga in his first season in Germany and was a regular this term as they finished in mid-table.
Wimmer is Spurs’ first close-season signing, but Sky sources understand a £12m bid is being prepared for Burnley striker Danny Ings.
Wimmer is now looking forward to the challenge of breaking into Tottenham’s first team.
He told Cologne’s website: “Tottenham is a huge chance for me. Nonetheless, it won’t be easy saying good bye to Cologne.
“I experienced wonderful moments here, such as the promotion and the Klassenerhalt (staying up).
“I want to thank the training staff, my team-mates, and all those involved in the club for their extraordinary support.
“Cologne and its fantastic fans will always remain something special for me.”
Cologne general manager Jorg Schmadtke added: “Kevin has improved tremendously during his time here and has developed into an Austria international player.
“Of course it is with some regret that he will be leaving us, but we understand that he is getting the opportunity to play for one of the biggest traditional clubs in London.
“In our dealings with Tottenham we were able to reach an agreement that all parties were happy with.”