The German is entering the last 18 months of his current deal with the Bundesliga side and Leipzig chief executive Oliver Mintzlaff wants him to commit to a new contract.
Werner is a key target for the majority of Europe's elite sides with Paris Saint-Germain and Barcelona also said to be keen on adding the pacey forward to their attacks.
However, according to German newspaper Bild, Leipzig have now offered to double the 22-year-old's salary worth £5.2million-per-year in a bid to keep their main main.
His new deal would also feature a £60m release clause that 10 leading clubs would have to pay to enable them to sign him.