The Belgium international, 25, completed a medical on Monday and Swansea say they can make the move permanent at any stage of his lengthy loan.
Lamah has made 15 appearances in La Liga this season, scoring twice and claiming two assists, and says he is excited by the chance to join Swansea’s Iberian-inspired squad, and work under former Barcelona and Real Madrid star Michael Laudrup.
“This is a great opportunity for me and I’m happy to be here,” said Lamah. “I know a few of the players here, and I already know that Swansea is a family club that will suit my style of play.
“To play in the Premier League is a dream, and also to play under Michael Laudrup is fantastic for me. I am looking forward to starting training and meeting the squad, and then I want to show everyone what I can do.”
Lamah began his career in his native Belgium with Vise before joining Anderlecht for four years, the last of which was spent on loan at Roda JC in the Netherlands.
He subsequently joined Le Mans in France before Osasuna signed him for £1.25 million in the summer of 2011.
“We’ve been watching him for a while,’’ said Swansea chairman Huw Jenkins. “Michael (Laudrup) has been looking to strengthen the wide positions to create more competition and we are delighted everything has been agreed.
“He is a player with slightly different attributes to what we already have. Hopefully he will settle in quickly and perform to his best.’’