Bayer Leverkusen are keen to have a new left-back in place before their training camp begins on Saturday, having lost Alejandro Grimaldo to Atlético Madrid this summer.
Gutiérrez, who joined Napoli from Girona for around €18m last summer and has a contract in Naples running until 2030, would represent a rapid swoop should the German club move quickly enough.
It remains unclear whether Facundo Medina, the Argentina international with whom Bayer are also negotiating, would be earmarked for the left side of defence or a more central role, adding a further layer of uncertainty to Leverkusen's defensive rebuild as they try to replace last season's key contributor at left-back.
Also to be resolved by Saturday, according to Bild, is the situation surrounding 18-year-old Kerim Alajbegovic, though the report gives no further detail on what that decision might involve.
With the clock ticking down to the start of pre-season training, Leverkusen's business over the coming days looks set to shape the composition of their backline heading into the new campaign, with Gutiérrez now firmly in the mix alongside Medina as the club works to find a long-term successor to Grimaldo before deadline pressures start to bite.
Gutiérrez came through Real Madrid's academy before making his name at Girona, where he spent three seasons and played a key role in the club's historic Champions League qualification, catching the eye of several of Europe's biggest clubs before his move to Serie A champions Napoli last summer.
He has struggled for regular game time under Antonio Conte in Naples, making him a realistic target for a club like Leverkusen looking for a proven performer at a relatively modest fee compared to some of the other names still on the market at left-back this summer.