Spanish radio station COPE reported on Monday that Atletico Madrid regard selling Álvarez to Barcelona as a "matter of honour" after the Catalan club's handling of the situation, and have made clear they will not negotiate with their La Liga rivals regardless of the fee offered.
The Argentine forward confirmed his desire to leave Atlético after the club's 2-0 World Cup victory over Austria, in which Lionel Messi scored both goals.
"I don't think this is the right moment to talk, but I don't want to hide either," Álvarez told ESPN. "I spoke with the people at Atlético that I needed to speak with.
"I think the best thing for everyone is a transfer. I want to fulfil my dream."
Multiple reports indicate that dream is a move to Barcelona, and Álvarez has reportedly refused to engage with transfers that do not align with that wish. Atlético, however, have signalled they are prepared to keep him out of the squad entirely rather than sell to their domestic rivals.
Instead, Atlético are asking Arsenal for between €40m and €60m in cash plus Gyökeres, 28, in exchange for the 26-year-old. Diego Simeone is said to want a straightforward number nine as Álvarez's replacement, and the Sweden international — signed from Sporting CP last summer — fits that profile.
Arsenal have long been linked with Álvarez and lines of communication remain open. Reports suggest the Gunners would be reluctant to include Gyökeres in any deal, however, with manager Mikel Arteta and sporting director Andrea Berta both believing the striker still has significant room for development at the Emirates Stadium.
A Gyökeres exit could in turn open the door for Alexander Sörloth's sale, given the similar profiles of the two forwards.
Álvarez and Gyökeres are both contracted to their current clubs until 2030.