The 20-year-old signed for RB Leipzig from Championship side Nottingham Forest last summer for £13m but struggled in his debut season.
Although his side are in the Champions League this year after finishing runners-up in the Bundesliga it is understood that Burke is open to a return to England, with a number of Premier League clubs monitoring the situation.
Crystal Palace are one of the clubs expressing an interest following Wilfried Zaha's knee injury which could leave the winger out for more than four weeks.
The Scotland international is also being monitored by Bournemouth, Chelsea and Tottenham.
New Palace boss Frank de Boer is looking to bolster his attacking options following a nightmare managerial debut where his side were thrashed 3-0 by newly-promoted Huddersfield.