The Bavarian giants have reportedly upped their bid for the Germany international, who is also attracting interest from Old Trafford with Edwin van der Sar set to retirehere.
Bayern Munich have increased their offer for Schalke's Manuel Neuer to €21 million (£18m), according to Bild.
The reported deal would include €18m (£15.8m) upfront with a further €3m (£2.6m) to be paid in the future. If the Gelsenkirchen outfit and Neuer, who has also been linked with Manchester United, agreed to the deal, the club could earn a further €7m (£6m) from performance-related bonuses.
The latest offer comes after Bild reported on April 27 that Bayern had an offer totalling €18m (£15.8m) rejected by Schalke.
It is not yet known whether the latest offer will be accepted by the Champions League semi-finalists' board but the club's sporting director Horst Heldt found Bayern's previous offer unacceptable.
"The offer from Bayern was not worth mentioning, not acceptable. It is seriously worth considering keeping the world's best goalkeeper for another year," the sporting director told German media last month.
However after Tuesday's negotiations, Heldt seemed to indicate that a deal may be close, telling Bild: "Yes, we [Schalke and Bayern] had a decent conversation."
Neuer is likely to play his final Bundesliga game for Schalke at the weekend when his side travel to Koln.