Belgium international Lamah joins on an 18-month loan deal from Osasuna, subject to Premier League clearance.
The Swans have the option of turning the move permanent for the 25-year-old winger at any time during the loan spell.
"This is a great opportunity for me and I'm happy to be here," said Lamah to the Swans website. "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."