The comments reinforced a position he first set out publicly on 17 July and effectively closed down weeks of speculation linking the Argentine with a move to Barcelona.
"Atlético Madrid, like all the great clubs, is built with great players, and I'm clear that Julián is one of them, as well as being a great professional," the Atlético chief executive said.
"We have to compete with our values, with people from the club and with the best players we can bring in from outside."
Gil said the club intended to support Álvarez through what has become a turbulent summer, stressing his importance to Atlético's wider ambitions.
"We're going to help Julián and continue looking after him so he can give us his full potential, because we need him to keep competing among Europe's elite," he said. "We have to understand that Julián hasn't had the best advisors around him regarding his professional career in recent months."
The Atletico Madrid chief executive also directed a pointed message at Barcelona and the outlets he believes have amplified the saga.
"We can't allow other clubs and their affiliated media to decide what we have to do, dictate our sporting strategy and disrespect us," he said. "That's exactly what they're looking for: to divide us. It's clear we're not as powerful as they are, but we have values. Atlético fans won't allow ourselves to be contaminated. Atleti is strong when we're all together."
Atlético's position has not shifted since the club first addressed the speculation months ago: Álvarez remains under contract, continues training normally under Diego Simeone, and could be included in the squad for Sunday's match against Villarreal. The club points to his contract and release clause as ruling out any departure, whether to Barcelona or Arsenal, who have also been linked with the striker.
The forward's representative, Fernando Hidalgo, responded within hours on social media, appearing to take direct aim at Gil's suggestion that Álvarez had been poorly advised.
"Never ask a liar why they lied," Hidalgo wrote. "To explain themselves, they'd have to lie again."
The exchange adds a further chapter to a saga that has dominated Spanish football coverage for much of the summer, with Álvarez having previously spoken publicly himself about his interest in leaving Atlético for Barcelona.
Whether that ambition can be reconciled with a club insisting there is nothing left to discuss remains to be seen, though for now Atlético's position leaves no obvious route for a move to materialise before the window closes.