The Atletico Madrid forward has attracted strong interest from a cluster of elite clubs, including Barcelona, Paris Saint-Germain and Chelsea.
Alvarez was on target against Arsenal on Wednesday night, converting a penalty in a 1-1 Champions League semi-final first-leg draw.
He will now look to knock the Gunners out in the return leg next week, all while aware that the Emirates Stadium could yet become his next destination.
Discussions over a potential move have been ongoing since last year, when Alvarez began to think about the prospect of a transfer.
Barcelona have been suggested in reports as his preferred option, and the Catalan club have been in active talks with his camp. However, they face financial limitations that could complicate any deal, sources confirmed.
It means a move away from Spain is a very real possibility for the 26-year-old. Paris Saint-Germain and Arsenal have both remained firmly in the race, but Chelsea could come into contention too.
Former Manchester City forward Alvarez is believed to be open to the prospect of one day returning to England in the right conditions.
Arsenal would need to assure the player of his role in the side, particularly given their current set-up and the presence of Viktor Gyokeres in the centre forward position.
Mikel Arteta is a big admirer of the player and has been told the board will back him for further signings in the summer window.
Atletico Madrid would ideally like to keep Alvarez, but accept that a sale may be necessary. Their valuation is going to be in excess of €100 million, but negotiations could bring the price down.
A move to La Liga rivals Barca could directly impact Atletico more dramatically this summer - and with that in mind, Arsenal sources believe the club could find themselves in a good position to sign him.