Laudrup would become the fourth manager in five years for the Welsh club and will be looking to build upon their successful first season in the Premier League in which they finished 11th.
Laudrup enjoyed a distinguished playing career, starting with Dansih club KB and finishing in 1998 at Ajax.
In between his time with those clubs, he played for both Barcelona and Real Madrid where he won successive La Liga titles playing with the two Spanish giants.
As a manger he tasted success with Danish club Brøndby, capturing a Danish Cup in his first year in charge while leading the club to a domestic double in 2005.
Following his departure from Brøndby in 2006, he took charge of Spanish side Getafe, guiding them to a Copa Del Rey final and the UEFA Cup quarterfinals.
He hasn't taken charge of a team since his last game as manager of financially stricken Mallorca in 2011.
As a player and manager, the two-time Danish Player of the Year promoted a youthful and attractive brand of football.
This attitude will be comforting to Swansea's fans after their side had one of the highest pass completion rates in Europe in 2011/12.
It is expected that Swansea chairman Huw Jenkins will make the announcement before the end of the week.