The Bundesliga champions are said to have tabled a bid worth €32 million for the 21-year-old, although Brazilian side Santos are unwilling to negotiate unless that figure rises substantially.
According to RTI Esporte, Santos value the midfielder at €49 million and have informed Bayern that talks will only progress if the German club improves its proposal.
Despite rejecting the initial offer, Santos are reportedly open to selling Bontempo during the current transfer window because of the club's financial situation. The report claims the Brazilian outfit need to generate funds, making a transfer increasingly likely if Bayern return with an improved bid.
Bayern Munich's interest comes as Vincent Kompany's side continue to reshape their midfield options ahead of the new campaign.
Young midfielder Noel Aseko is expected to complete a move to Eintracht Frankfurt, while João Palhinha has been linked with a departure only a year after arriving in Munich. Those developments have prompted Bayern to assess further options capable of strengthening the centre of the pitch.
Bontempo has emerged as one of South America's most highly rated young midfielders thanks to his technical quality and versatility. Comfortable in a variety of central roles, he has established himself in Santos' first team and is viewed as a player capable of making the step up to European football.
Whether Bayern are prepared to meet Santos' asking price remains uncertain.
The German champions have already invested heavily in strengthening their squad this summer, with left-back Nathaniel Brown and attacking midfielder Ismael Saibari among the club's notable arrivals.
For now, negotiations appear to hinge on Bayern's willingness to increase their offer closer to Santos' valuation.
Neither club has commented publicly on the reported bid, while discussions are expected to continue if Bayern decide to return with an improved proposal.