The Welsh side have won just once in their past seven matches and are 17th in the Premier League table.
Swansea’s American owners, Steve Kaplan and Jason Levien, and chairman Huw Jenkins ended the 10-month tenure of Italian Guidolin on Monday with the club above the relegation places only on goal difference.
Bradley beat ex-Manchester United assistant manager Ryan Giggs, former Derby boss Paul Clement and the one-time Villarreal manager Marcelino to the job after spells in Egypt, Norway and France since leaving the USA in 2011.
He spent last season at French second-tier side Le Havre, missing out on promotion to Ligue 1 by a single goal.
“I am sorry to leave, especially at this moment,” said Bradley after Monday’s 2-1 win over Sochaux, a result that left Le Havre fifth in Ligue 2.
“In a perfect world I would go right to the end with this team just like last year – but football does not control timing.”
Bradley is expected to take his first training session at Swansea on Wednesday, with the club’s next game against Arsenal at Emirates Stadium on Saturday, 15 October after the international break.
His first decision is likely to be the make-up of his support staff.
Guidolin’s backroom team – Diego Bortoluzzi, Gabriele Ambrosetti and Claudio Bordon – were released alongside the 61-year-old, who was dismissed on his birthday.
Club stalwart Alan Curtis, who spent time as interim manager last season following Garry Monk’s December departure, is expected to play a key role at the Liberty Stadium.