Reports on Friday claimed that the Gunners had expressed a concrete interest in signing the 30-year-old and had already opened talks with his entourage over a free transfer.
Kimmich entered the final six months of his contract at the Allianz Arena in January, meaning that Bayern are powerless to prevent him from entering discussions with foreign teams over a pre-contract agreement.
At one stage, the Germany international was allegedly close to sealing a three-year extension at Bayern Munich until 2028, only for the Bundesliga leaders to withdraw their offer due to the player hesitating.
The 30-year-old supposedly thought twice about putting pen to paper due to one proposal from one Premier League side and one team from abroad, but Bayern are not prepared to wait around much longer.
Kimmich 'given until Wednesday' to decide future
According to Bild, Bayern have given Kimmich until next Wednesday - the date of their Champions League last-16 first leg with Bayer Leverkusen - to come forward and indicate his willingness to sign a new deal.
The former Stuttgart man should not expect a salary increase on any new terms following his initial hesitation, though, and Bayern will apparently now only sanction an extension for the same amount or less money than he is currently on.
The report adds that Kimmich would have been content to continue on his current wage, but Bayern bosses were left dissatisfied with his demands over a signing bonus, another reason why they pulled the plug.
Amid the uncertainty over his future, Kimmich has remained a vital part of the Bavarians' setup this season, scoring one goal and providing 10 assists from 36 matches across all competitions.
The 30-year-old has amassed 43 strikes and 114 assists in 426 matches for the Bundesliga leaders since joining in 2015, but he missed Friday's 3-1 win over Stuttgart due to a tendon injury.
Which other clubs are interested in signing Kimmich?
Arsenal's interest in Kimmich appears to be the most concrete at the minute, as Mikel Arteta eyes a cost-effective deal for a player with trophy-winning experience and who could replace two potential wantaway midfielders.
Thomas Partey and Jorginho are both expected to depart when their deals run out at the end of the campaign, although the Gunners are believed to be pressing ahead with a £51m deal for Real Sociedad's Martin Zubimendi as well as Kimmich.
While the Germany international would likely be utilised as a number six at Arsenal, both Manchester City and Liverpool are thought to be eyeing him as a new right-back if their long-serving options leave.
City are known to be prioritising a fresh face in that position as Kyle Walker prepares for a permanent exit, while Liverpool will scope out a replacement for Trent Alexander-Arnold if he is seduced by Real Madrid.
Alternatively, Los Blancos could target Kimmich if Alexander-Arnold elects to stay at Liverpool, while Barcelona have also been credited with an interest.
Written by
Ben Knapton