The 37-year-old is 6-10 with a 4.15 ERA and 123 strikeouts in 149.2 innings this season for the Phillies.
Burnett's current contract runs through the 2014 season, but he does have a mutual/player option for 2015 tied to his contract.
Rosenthal also notes that Royals pitching coach Dave Eiland worked with Burnett during the pair's time with the New York Yankees.
The Royals are 54-52, five games back of the American League Central-leading Detroit Tigers. The team is also 3.5 games out of the second American League wild card spot.
- Sarah Barshop