Flick doesn't want Barça to spend the entire summer saving money for a deal that may never have had a realistic chance of going through.
That's why he has asked Deco to let him know the Argentine's true intentions as soon as possible," the report stated.
Concerns remain within the Catalan club that their public interest is merely being utilized as tactical leverage in parallel negotiations.
"The fear is that Barcelona's interest will ultimately serve only as a negotiating tool.
If Arsenal needs to increase their salary offer to convince Julián, or if Atlético Madrid wants to secure him with an even more lucrative contract extension, being able to mention Barça as an alternative gives the player a strong position during talks with either club," the report added.
Former Manchester United midfielder Paul Scholes advocated for Arsenal to finalize a deal for the World Cup winner.
"There's not even a debate. I don't think there's loads of top-class centre-forwards knocking about.
I don't see him as an out-and-out centre-forward, he's more of a drifter," Scholes said.
Scholes highlighted the competitive edge Alvarez would bring to the Premier League side given current market valuations.
"Atletico Madrid won't want to sell him to Real Madrid or Barcelona.
If he's £100m – [England midfielder] Elliot Anderson has just gone for £116m – and Alvarez is a match-winner. What a signing that would be," Scholes added.
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Meanwhile, reports from Mundo Deportivo suggest Barcelona retains a 100 million euro bid while monitoring Arsenal's separate interest in Real Madrid winger Vinicius Junior, which could shift Arsenal's focus away from Alvarez.