The 20-year-old spent the second half of last season on loan at the German club, and the Independent now report that he could be set to return to Germany on a permanent basis.
Leipzig have already had two offers - worth around £12m and £14m - rejected for the England Under-21 international, with Everton said to want £20m for their young talent.
With the Toffees braced to receive a third bid, it seems likely that Leipzig will push closer to a figure that will be deemed acceptable by Everton.
A sale at their asking price would make Everton a £12m profit on the fee they paid Everton for the youngster in January 2017.
During his half a season in the Bundesliga, Lookman scored five goals as he helped the side qualify for next season's Europa League.
Lookman himself is said to be keen on a move to Germany, as he would be valued as a key player by the club - something that is not so likely to be the case at Everton.
Marco Silva is still assessing the Everton squad he has inherited over the summer, and could see Lookman as a young fringe player with potential - something the 20-year-old may not settle for.
Lookman's Everton future could be plunged into further doubt if the club manage to seal moves for Malcom and Wilfred Zaha - with whom they have been linked - as they are both players that can play in Lookman's position.