"In this moment, the opportunity to go to the Premier League on many levels is special," he said after leaving French second tier club Le Havre.
"I am sorry to leave, especially at this moment, but it's still an opportunity for me, for my family and for American football. That's important," the 58-year-old added.
"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 has been tasked with improving a struggling Swansea side, who are fourth-bottom of the table after seven games.
His first game in charge is against third-placed Arsenal on Oct. 15.
(Reporting by Shravanth Vijayakumar in Bengaluru, editing by Ed Osmond)
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