The former Barcelona and Real Madrid midfielder, who had previous managerial experience at Bronby, Getafe, Spartak Moscow and, most recently, Real Mallorca, has signed a two-year contract at the Liberty Stadium.
The announcement follows the departure of previous boss Brendan Rodgers to Liverpool at the beginning of June, and an officially unveiling in expected next week.
The Northern Irishman took the Swans to the Premier League for the first time in their history last summer, before guiding them to a highly-respectable 11th-place finish.
Laudrup, has previously spoken of his desire to manage in the Premier League, enjoyed much success while in charge of Bronby, overseeing four domestic cup triumphs and one league title, including a treble-winning season in 2004-05.
Since then, however, the 48-year-old has struggled to hold down a permanent position; he resigned after a short spells in charge of Getafe and Real Mallorca, and was sacked from his position in charge of Spartak Moscow.
Laudrup's tactical approach is thought to be in keeping with chairman Huw Jenkins' ideals, following the original conception at Swansea by Roberto Martinez.
Marcel Desailly was approached for the position but turned down the opportunity has he had only just started his Uefa coaching licence.
Blackpool manager Ian Holloway and Brighton's Gus Poyet were all also considered, along with their former assistant manager Graeme Jones.