A fee for the 21-year-old frontman has been agreed, with Garry Monk eager to bolster his attacking options at the Liberty Stadium.
Castaignos is yet to arrive in Wales, though, and Swansea are continuing to explore other options.
Twente claim there is no cause for alarm on the transfer front, with there still every chance that a summer switch will be pushed through.
They fully expect discussions to be re-opened once Castaignos returns from a well-earned break, with Swansea chairman Huw Jenkins also currently out of the country.
Twente chairman Joop Munsterman told Voetbal International: “It will be a while before things can be done.
“The choice is up to the player. Only when Luc is back from vacation and is negotiating with Swansea can we expect progress. And as I have said, the clubs can work it out.”
Former Inter Milan striker Castaignos netted 14 times in the Eredivisie last season and boasts a strike rate of close to a goal every other game since returning to his native Holland in 2012.