The deal to bring in Kvist, who has also been linked with a move to West Ham United, will include an option to purchase the player for £1.25m in the summer.
"Midfielder William #Kvist from @VfB Stuttgart will join @FulhamFC on a loan," Bild said in a tweet. "In [the] summer Fulham can buy him for £1.25m. #vfb #ffc."
Stuttgart head coach Thomas Schneider also confirmed that the player was likely to leave the club this January.
"William Kvist feels unsettled at the club; he is looking for a new sort of competition," he told Stuttgarter Zeitung. "William has not been a first-team regular recently and, of course, as he is a Danish international, he requires to play regular football."
Kvist has found himself on the fringes of the Stuttgart first team this season.