The 24-year-old has recently emerged as a potential target for Thomas Tuchel's side, who are looking to bolster their midfield options before the transfer window shuts at 11pm on Thursday.
A recent report claims that the West Londoners have made a €50m (£43m) bid to sign Alvarez, who still has three years remaining on his contract in Amsterdam.
According to Dutch news outlet De Telegraaf, the Mexico international is refusing to train and wants Ajax to enter negotiations with Chelsea over a late deadline day move.
The report adds that Alvarez is determined to push through a transfer to Stamford Bridge and he is now the third Ajax player to request a transfer this summer after attacker Antony - who has now joined Manchester United - and midfielder Mohammed Kudus, who has attracted interest from Everton.
Ajax have already recouped around £180m from player sales this summer, with Antony, Lisandro Martinez, Sebastien Haller, Ryan Gravenberch, Perr Schuurs, Nicolas Tagliafico and Dominik Kotarski all leaving the club on permanent deals.
The Eredivisie champions have no interest in losing any more of their star players in the final hours of the transfer window, especially now that they are unable to recruits replacements as the Dutch transfer window closed on Wednesday night.
Both Rennes and Newcastle United are understood to have seen bids for Alvarez turned down by Ajax earlier in the window, but the versatile midfielder has his heart set on a switch to Chelsea, who are awaiting a response from the Dutch club regarding their offer.
Since joining from Club America in 2019, Alvarez has made 107 appearances for Ajax across all competitions, winning two Eredivisie titles and one KNVB Beker.
The 58-cap Mexico international predominantly operates as a defensive midfielder, but can also play at centre-back, and has done so on several occasions for both club and country.
Alvarez has been identified by Chelsea as a potential midfield target, with Tuchel looking to bolster his options in the middle of the pitch to ensure they have enough strength in depth to compete on all fronts.
Midfield duo N'Golo Kante and Ruben Loftus-Cheek are both on the sidelines with hamstring injuries at present, with the latter forced off at half time in Chelsea's 2-1 defeat away at Southampton on Tuesday night.
Mateo Kovacic is another who has struggled to gain match fitness after a spell out with a knee problem and he is yet to start in any of Chelsea's opening five Premier League matches this term.
Meanwhile, the futures of Billy Gilmour and Conor Gallagher have been questioned, as both players have been linked with permanent moves away from Stamford Bridge.
Alvarez is not the only player Chelsea are targeting on deadline day, as they have reportedly reached an agreement with Barcelona to sign forward Pierre-Emerick Aubameyang for €14m (£12.1m).