The RB Leipzig attacker has been heavily linked with both clubs in recent months and even admitted back in December that he would consider a switch to the Allianz Arena.
Werner has been offered a new deal at the Red Bull Arena but he is stalling on signing an extension with the club, with his current contract set to expire in 2020.
The 22-year-old has enjoyed another impressive season at Leipzig, contributing 14 goals and two assists in 25 matches across all competitions.
At international level Werner has also become an integral part of Germany's starting XI, earning 23 caps and finding the back of the net on nine occasions.
Should he fail to sign a new contract at Leipzig the club are ready to cash in on their talismanic striker this summer, but Mateschitz has revealed that only lucrative offers will be considered.
When asked about the possibility of Werner being sold when the transfer market reopens Mateschitz told German publication Blick : "For sure that player [Werner] will not be cheap."
The German star is currently valued at around £50m ($66m) and a number of other top European clubs have reportedly expressed an interest in him recently, including Real Madrid, Atletico Madrid, Manchester United and Arsenal.
However, Liverpool and Bayern are thought to the frontrunners in the race to land Werner's signature, with Sport Bild going as far as suggesting an agreement is already in place with the Munich giants.
Reds midfielder Naby Keita has already tried to persuade his former team-mate to complete a switch to Anfield, while Sadio Mane has previously stated: "He would be welcome in our team".
Liverpool, who are locked in a Premier League title battle with Manchester City this season, could well look at signing Werner to help boost an already strong attack.
Bayern, meanwhile, have once again relied on Robert Lewandowski as their main source of goals although could well have identified Werner as his long-term replacement.