According to multiple German media reports, former Bayern Munich defender Martin Demichelis, 45, is the leading candidate to replace him.
Werner, 38, took charge of Leipzig last summer on a contract running until 2027, arriving from Werder Bremen after guiding them for 128 matches.
He led RB Leipzig to third place in the Bundesliga this season, finishing with 65 points, just two short of the club's own record tally.
That result secured Champions League qualification for next season.
Despite those results, reports suggest there has been internal dissatisfaction with Werner's style of play and with how narrowly Champions League qualification was achieved.
According to Sky Sport Germany, sources point to unhappiness from supervisory board chairman Oliver Mintzlaff and the club's global football department, which includes Jurgen Klopp and Mario Gomez.
Demichelis, who currently manages RCD Mallorca, took charge of the Spanish club in February on a deal running through to 2028.
Despite that contract, reports out of Spain suggest he is ready to leave Mallorca for Leipzig, with Spanish outlet Marca describing the move as effectively agreed.
South American broadcaster BolaVip reported that Red Bull are prepared to pay a compensation fee of around €3m to secure his services.
As a player, Demichelis spent time at both Manchester City and Bayern Munich during a career mostly spent at centre-back.
Werner was not Leipzig's first choice when they appointed him last summer.
The club had reportedly tried to land Cesc Fabregas, who instead remained at Como and led the Italian club into the Champions League.
Leipzig's chief executive Marcel Schafer had previously indicated a willingness to discuss a new contract with Werner following the club's Champions League qualification.
However, by early June he had begun to row back publicly, citing an ongoing end-of-season review.
That review now appears to have concluded, with reports suggesting an outcome unfavourable to Werner.
Leipzig enter the new season having already lost key attacking players, with Xavi Simons, Loic Openda and Benjamin Sesko all departing last summer.
A managerial change so soon after Werner's arrival would mark another reset for a club that has cycled through several coaches in recent seasons.
For Demichelis, a move to Leipzig would represent a significant step up from Mallorca and a return to one of Europe's higher-profile coaching positions.
Nothing has yet been confirmed by either club, but the volume and consistency of reporting from Germany and Spain suggests an announcement may not be far away.