CBS Sports' Jon Heyman.
Swisher is considered one of the bigger targets in a very weak free agent market this offseason that includes the likes of Josh Hamilton and B.J. Upton, among others.
Heyman is reporting that the Mariners, Orioles, Phillies, Braves, Nationals and Giants have shown interest in recent weeks.
Swisher is expected to receive around $12 million per year for three to four years and has already turned down a one-year offer from the Yankees.
Swisher hit .272 with 93 RBI and 24 home runs in 2012 in the hitter-friendly new Yankee Stadium.
He played for the Oakland Athletics from 2004-2007.