Lookman spent the second half of last season on loan with the Bundesliga team, who are keen to secure a permanent deal for the Englishman, making 11 appearances in which he scored five times and created a further four goals for his teammates.
However, according to Bild, their two bids of €14m and €16m have both been rejected by Everton, who are holding out for €20m, around £18m.
Ralf Rangnick, manager of RB Leipzig, has insisted that the German side are looking to secure the permanent transfer of Lookman, quoted by the newspaper as saying: "Ademola Lookman is currently a player of Everton. He knows from his time with us and also through the relationships he has built up with us, our appreciation for his quality as a player and as a person.
"We are in constant contact with his club."
However, RB Leipzig are not prepared to meet Everton's valuation of Lookman, and face a battle with the Toffees over Lookman's future. Bild claim that Lookman has expressed his desire to return to the Bundesliga side, but so far no deal has been agreed between the two parties.
Lookman is under contract with Everton until 2021, meaning they have no urgent need to sell the 20-year-old forward. He was initially signed from Charlton Athletic in January 2015 for a fee of around £8m, and the club are now seemingly looking to earn twice as much from Lookman's potential departure.
New manager Marco Silva is believed to be very keen on Lookman, and is planning for next season with Lookman as a key member of his squad.
During Everton's pre-season friendly against ATV Irdning, who are in the fifth tier of Austrian football, Lookman netted a hat-trick in the first half of the match, which ended up finishing 22-0 to the Premier League side.