Lookman, 23, has often dazzled in his loan spell from Germany.
Now Cottagers boss Scott Parker would like him to stay if that can be arranged.
However, Leipzig still want a sky-high fee and that cannot be easily arranged with Fulham in their current league position.
They will struggle to commit to a signing at this point but plan to try if they stay up.
Former Everton winger Lookman has scored twice, adding three assists, in 14 Premier League games this season.