The Toffees' director of football Marcel Brands and manager Marco Silva have remained steadfast in their refusal to sell, with Leipzig's latest offer of £22 million ($28.3m) - an amount which would have broken their transfer record - rejected by the Goodison Park outfit.
Former Charlton Athletic starlet Lookman spent the second half of last season on loan at Leipzig, with a series of impressive performances leading to five goals and four assists in 11 Bundesliga outings.
Such form has unsurprisingly seen Leipzig retain their interest throughout the transfer window, with their latest bid suggesting they are willing pay more for Lookman than they did to Red Bull Salzburg for current record purchase Naby Keita in 2016.
Lookman, 20, is seen as one of the brighest young talents in England having helped the Three Lions to win the Under-20 World Cup in 2017, though he has struggled for regular game time with Everton.
Since arriving at the club in a move worth up to £11m ($14.1m) from Charlton in January 2017 he has made just 24 first-team appearances, totalling 878 minutes.
Silva has suggested previously Lookman would be open to a move back to Germany, though any deal would need to be sealed by August 31 when the transfer window in Germany closes.
Everton are well stocked in wide areas following the arrival of Richarlison from Watford over the summer, though Yannick Bolasie has reportedly held talks with Middlesbrough and Aston Villa regarding a loan move to the Championship.