Messi became a free agent when his Barcelona contract expired on June 30, and planned to stay until the club announced that financial problems would prevent them from registering a new, agreed deal.
The 34-year-old eventually signed for Paris Saint-Germain on August 10, signing a two-year contract with the option of a further year.
"When what happened at Barcelona happened, we called Luis," Simeone told Argentinian newspaper Diario Ole on Tuesday. "I didn't call Leo. But I called Luis to say 'look, how is he? Would he be keen? Is there the slightest possibility... imagining that he could come to Atletico Madrid."
Suarez -- Messi's closest friend and confidant in their time together at Barcelona -- had moved to Atletico a year earlier in September 2020, after being told he was surplus to requirements at Camp Nou by coach Ronald Koeman.
The Uruguay forward has enjoyed a career renaissance in Madrid, scoring 21 league goals last season to help fire Atletico to their first LaLiga title since 2014.
Simeone said that his hopes of landing Messi were short-lived, once it became apparent that talks with PSG had already progressed significantly.
"It lasted about three hours," he said. "Obviously Paris Saint-Germain were obsessed with that signing. The truth is that [coaching Messi] hasn't happened because he's been at Barcelona, we've been at Atletico, and while he's been with the national team I haven't been there."
He added, laughing: "It's like an aeroplane that you see in the sky and say 'there it goes...'"
Messi scored his first goal for PSG last month, in a 2-0 Champions League win over Manchester City.
"If you ask me where Messi should play: in a team that wants to win," Simeone said. "He should play in a team that knows what to do to win. It doesn't matter where. In a team that wants to win and is prepared to win. Don't worry about him. Worry about the team."