The former Derby County boss is thought to have agreed a two-and-a-half year deal to take over the Premier League strugglers following the sacking of Bob Bradley last month.
Bayern chief executive Karl-Heinz Rummenigge has revealed that Clement "expressed a desire" to return to English football and was therefore granted the chance to discuss terms with Swansea, with the German giants moving quickly to bring in a new replacement.
"Paul has been a professional, loyal and capable coach for us," Rummenigge told the club's official website. "We did not want to stand in his way of becoming a manager in the Premier League. We wish him all the best for the future."