The German newspaper described Klopp, Red Bull's global head of football, as the central figure pushing for the change.
Werner has not yet been officially dismissed, but reports in both Germany and Spain suggest his departure is close.
According to Bild, Klopp's relationship with Werner has cooled despite Leipzig returning to the Champions League under his management this season. The newspaper reported that Klopp's connection with Demichelis runs through a shared country club in Mallorca, where the two men are said to mix socially.
Demichelis, a former Bayern Munich defender, has been in charge of RCD Mallorca in LaLiga.
He is reported to be available to Leipzig for a fixed fee of €2.5m, with Mallorca said to be willing to let him leave via a release clause. Werner only took charge at Leipzig last summer, arriving after declining to extend his contract at Werder Bremen.
He guided the club to third place in the Bundesliga and secured a return to the Champions League, missing the club's own points record by just two.
Despite that, reports in Germany suggest senior figures at Red Bull's global football operation, including supervisory board chairman Oliver Mintzlaff, had reservations about Werner regardless of the results.
Demichelis previously managed River Plate and Manchester City, where he worked as a coach under Pep Guardiola before taking over as head coach following Guardiola's departure from the role at the Argentine club. He recently extended his contract at Mallorca to 2028, but reportedly did so with an exit clause that would allow Red Bull's approach to proceed.
Werner's current deal at Leipzig runs until the summer of 2027.
For RB Leipzig, who came within touching distance of a club-record points total this season, the prospect of dismissing a coach who delivered Champions League football would represent a striking statement about where power genuinely sits within Red Bull's footballing structure.
Whether Werner's exit becomes official before pre-season begins may now depend on how quickly Demichelis can be freed from his obligations in Mallorca.