Swansea City were among those said to be tracking the 22-year-old as he prepared to leave Danish side AaB Aalborg.
The highly-rated winger has decided to shun a switch to Britain, though, and take on a new challenge in the Eredivisie.
He is convinced that he has made the right decision, with Heerenveen, PSV Eindhoven and Anderlecht also believed to have been in the running to secure his services.
Kusk told Tipsbladet.dk: “I checked the different options I had and it was Twente that were the best.
“It was a good match after my talks with Alfred (Schreuder, Twente coach).
“He knows what kind of player I am. He and Twente have got a plan for me and I know from (Andreas) Bjelland that it is a good club.”
Kusk, who has signed a four-year contract, added: “It is nice that it is finally sealed.
“It has been a longer process than I hoped and expected, but I am here now and I have already started training.
“The team has started with preparations for the new season and have three friendlies remaining, and that suits me fine as I know where I am with my own preparations.
“It suits me fine here and if I get a regular place, it will help my chances with the national team.
“I have to have patience, but I will try to deliver every day in training and earn a place as soon as possible.”