The Merseyside club ultimately opted for Alexander Isak and Hugo Ekitike rather than pursuing Alvarez. Yet questions persist over how much longer the Argentina ace will remain in the Spanish capital.
Speaking to Marca, Alvarez said: "Look, it doesn't bother me. I try not to pay much attention, but I know what's being said.
"It's all over social media. I think it's more what's being said online than what's actually happening.
"I'm very focused on this season, on what lies ahead with Atletico de Madrid. So, I try to tune it out and think about myself, about continuing to grow as a player, and about winning."
His representative, Fernando Hidalgo, also addressed the rumours after the summer window shut. He told Eram News: "No official offers were received from Liverpool to sign Julian last season.
"And everything that has been said about this is incorrect. During the last summer, there were no offers either, because Atletico Madrid did not want to sell the player, and therefore no negotiations were put on the table."
Come early October, Alvarez again touched on his situation when speaking to ESPN: "Look, I'm very calm about it all.
"People always talk. Last year, there was a lot of talk as well, but the season is just starting, and my focus is on getting better every day, doing my best for the club, helping my teammates and winning here.
"That's all I'm thinking about. What people say on social media is irrelevant."
Atletico's coach, Diego Simeone, had nothing but praise for Alvarez after his goal helped secure a 3-2 victory over PSV in midweek. He said: "Game-changing players like Julian have their moments, and they're brilliant.
"I hope he keeps playing like this because they're essential. Alex [Sorloth] showed great sportsmanship by setting him up for the first goal.
"Julian returned today, as many had hoped, to be the game-changing player he is. He worked hard, he ran, he had incredible individual moments."