The Swede's contract expires at the end of this month and he has been offered a bumper deal to return to his homeland.
Swedish top flight side Malmo want the right-back and have blown rivals AIK Stockholm out of the water with their offer, according to the Scottish Sun.
Lustig's Celtic future has been in doubt during the campaign but there was a belief that he may extend his stay for another season.
However, with no permanent manager in place Malmo could get the deal done early and prise the 32-year-old from Glasgow.
Lustig joined Celtic back in January 2012 from Rosenborg having signed a pre-contract deal.
He has become a decorated Parkhead icon, winning eight league titles, four league cups and heading into Saturday's Scottish Cup final showdown he sits on three Scottish Cups.
Lustig has endeared himself to the Celtic support over his time with his quirky antics and buying into the ehtos of the club.