Davies' current deal at Allianz Arena is due to expire next June, and both Manchester United and Real Madrid are believed to be battling to sign him on a free transfer.
Contract talks between Bayern and Davies are thought to have stalled in recent months, but according to Bild, the Munich giants have now placed a new deal on the table for the Canadian.
The report claims that Davies has told his Bayern teammates that he has been presented with a contract offer, with both Man United and Real Madrid seemingly suffering a blow in their pursuit of him.
Bayern sporting director Christoph Freund hinted on Friday that the left-back would pen a new deal at Allianz Arena.
"The confidence is there. It just makes me feel positive when I see him every day - how he comes from, how he plays, how much fun he has training and playing. This system, the way we play football at the moment, suits him extremely well. He feels extremely comfortable. He feels good in Munich and is playing at a really high level at the moment," Freund said.
Bayern 'offer in-demand Davies a new contract'
The 24-year-old has again been a key player for Bayern during the 2024-25 campaign, providing three assists in 14 appearances in all competitions, including one assist in eight Bundesliga outings.
Davies made the move to Allianz Arena from Vancouver Whitecaps in January 2019, and he has made 209 appearances for Bayern in all competitions, scoring 11 goals and registering 34 assists in the process.
Last month, the defender said that he 'felt valued' at Bayern and appeared to distance himself from rumours that he could leave the Bundesliga giants.
"As of now, my full focus is to stay healthy and to play the game with the club I'm with right now," Davies told reporters. "I'm not focused on what's going on in the future.
"I'm just focusing on staying healthy and helping my team win. My team make me feel valued. The whole Bayern community make me feel valued. I see that each and every day."
Will Man United, Real Madrid be forced to look elsewhere?
Man United's left-back pursuit will depend on Luke Shaw's fitness, as the defender is now back in training, and he will be desperate to put his injury problems behind him.
There are concerns when it comes to his long-term fitness, though, and a new left-back is expected to be a priority for the 20-time English champions if Shaw and Tyrell Malacia find it difficult to show their previous levels.
Bournemouth's Milos Kerkez is believed to be admired, with the 21-year-old again in impressive form for the Cherries this season, providing two assists in 10 Premier League appearances.
Ferland Mendy is currently Real Madrid's first-choice left-back, but the 29-year-old only has a contract at Bernabeu until next summer, and he could potentially be moved on despite his success at Bernabeu.