The decision effectively ends Arsenal's hopes of signing the forward this summer, despite the Premier League club continuing to push for a deal.
Alvarez has emerged as one of the most sought-after players in Atletico's squad. Real Madrid and Barcelona have also shown interest, with bids from both Spanish giants reportedly rejected by Atletico in recent months.
Barcelona are understood to be Alvarez's preferred destination should he leave Madrid. Atletico, however, have made it clear that they do not intend to allow the forward to join their domestic rivals.
That position had encouraged Arsenal to maintain their interest. The Gunners reportedly believed Atletico could be more willing to consider a transfer to a club outside Spain, potentially creating an opening for the Premier League side.
Atletico stand firm on Alvarez
According to Independent Sport, Atletico head coach Diego Simeone and the club's board have agreed that Alvarez will remain part of the squad this season.
The decision means Arsenal will not have an opportunity to complete a deal for the forward during the current summer transfer window.
Atletico's stance also represents a significant blow to Alvarez's hopes of securing a move. Although the attacker is understood to want a transfer, the Spanish club expect him to accept their decision and focus on the season ahead.
Alvarez has established himself among the leading attackers in world football since moving to Atletico and remains a key part of Simeone's plans.
Arsenal forced to look elsewhere
For Arsenal, the decision brings an end to their latest attempt to explore the possibility of signing Alvarez.
The Gunners have continued to monitor the situation despite Atletico's firm position, hoping that circumstances could eventually persuade the Spanish club to consider a sale.
That opportunity has now effectively disappeared.
Arsenal must therefore turn their attention to alternative attacking targets if they want to strengthen their forward options this summer.
With Atletico determined to retain Alvarez, neither Arsenal nor his other interested clubs will be able to negotiate a transfer before the window closes.
The forward is consequently set to remain in Madrid for the coming campaign, with Atletico hoping he can put the transfer speculation behind him and concentrate on delivering another strong season.
For Arsenal, meanwhile, the message from Atletico is clear: Alvarez is not available this summer.