The Sun report that Jurgen Klopp's old side are looking to move on from Swiss international Burki and view Mignolet as the man to replace him in goal.
30-year-old Mignolet signed for Liverpool for a fee of £9m in 2013 as a replacement for Pepe Reina and the Reds are hoping to make a small margin of profit off of the Belgian, valuing him at £10m.
A deal with the German outfit has also been made more likely following Stoke's relegation to the Championship this weekend. A rumoured swap deal with Jack Butland was thought to be in the works between Liverpool and Stoke, but that's unlikely to go ahead as Mignolet will likely be unwilling to play second tier football next season.
Following yet another disappointing season at Anfield, Mignolet has seen his Liverpool career stutter after being replaced by young German keeper Loris Karius.
Mignolet has made over 200 appearances in all competitions since his debut with Liverpool but has failed to win over the Anfield faithful, who have grown tired of his inconsistent performances and routine mistakes.
The rumours surrounding the 30-year-old's future coincide with comments made by former Liverpool man, Phil Thompson, concerning current number one Karius. Speaking to Sky Sports (via The Independent), Thompson suggested that Karius needs to be replaced this summer.
"I think most Liverpool fans will look at it and probably think we do need a new goalkeeper."
If a new goalkeeper is brought in then it will all but cement Mignolet's departure.