The highly-rated 21-year-old has drawn admiring glances from across Europe, with Bayern Munich another of those to have been linked with a big-money approach for his services.
Liverpool were reported to have identified the Germany international as a top target in the summer of 2017 before eventually being forced to look elsewhere and tie up a £36.9 million deal for Mohamed Salah.
Barca were also keen at that stage, as they sought to find suitable cover for the departed Neymar, but Brandt stayed put then and looks like doing the same during the winter window.
His father and agent, Jurgen, told Sport Bild of the ongoing transfer talk: “Julian's objective is to play in Europe with Leverkusen next season.
“He'll definitely be here until 2019. Julian feels he has unfinished business at Leverkusen.”
Brandt is only contracted to Leverkusen until the end of the 2018-19 campaign.
It is also reported that the terms of that agreement will see his asking price lowered from this summer, which is likely to ramp up the interest in his signature.
Leverkusen, though, are flying high in the Bundesliga table at present, with only runaway leaders Bayern ahead of them.
Brandt has contributed four goals and two assists to their top-flight cause, with a five-goal haul across all competitions having already seen him better his return from 2017-18.
His value to Leverkusen remains without question and they are understandably keen to retain his services for as long as possible, with it possible that fresh terms could be put in place to help fend off an every-growing list of suitors.
General manager Jonas Boldt has said on talks with Brandt: “We think that there's every chance he'll extend.”