The 58-year-old American replaced Francesco Guidolin at the Liberty Stadium after the Swans sacked the Italian on Monday.
Former USA national team boss Bradley led Le Havre to a fourth-placed finish in France's Ligue 2 last season, almost to promotion, and had the Normandy club sitting fifth on the table this term but has now opted to make the move to the Premier League which was too good an opportunity to pass up.
"To go to the Premier League is special," he said.
"It's still an opportunity for me, for my family and for American football. That's important.
"I am sorry to leave, especially at this moment.
"In a perfect world I would go right to the end with this team just like last year - but football does not control timing."