Lamah was born in the Ivory Coast but moved to Belgium aged 15, and has been capped five times by their national side since making his debut in 2009.
He has already played football in his adopted country, the Netherlands and France, and is looking forward to playing in England after completing a medical.
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“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.”
Swans chairman Huw Jenkins has also expressed his delight at the club making their first signing of the January transfer window.
“We’ve been watching him for a while,’’ he said. “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.’’
Lamah is ineligible for Swansea's FA Cup third round replay against Arsenal as he was not registered in time for the original tie, but will be available for the Swans' match at home to Stoke on Saturday