Messi has since ended a 21-year association with Barca as he joined Paris Saint-Germain on a free transfer, while Martinez recently renewed fresh terms with the Serie A side until the end of the 2025-2026 campaign.
The pair enjoyed success at the Copa America last July, collecting Argentina's first major trophy in 28 years as Messi starred with four goals.
But Martinez revealed the Blaugrana legend wanted to partner up with him at club level as well as he attempted to tee up a deal for the Inter man to join him at Camp Nou in 2020.
"We talked about it a lot in the national team, he asked me what the situation was like between me and Inter, but I always remained calm because we talked to the club to find an agreement to be happy here," Martinez told Sportmediaset.
"That's how it went and I'm happy to have stayed, I'm fine in Milan: I like my family."
Martinez, who signed from Racing Club in July 2018, was an integral part of Antonio Conte's title-winning team as he featured in every Serie A game for Inter.
The 24-year-old formed a dominant partnership last season with Romelu Lukaku, who left for former club Chelsea in August, but the Argentina international explained his strike partner's departure was a personal decision.
"Did he regret Chelsea? I don't know, he made a personal choice," Martinez added on Lukaku.
"He was happy at Inter, we talked often but if he chose to leave it was obviously because he felt like it. We are the ones who now work to give Inter the best."
Martinez, who netted 17 times last term, thrived under the now Tottenham head coach as Inter ended a 10-year wait for the Scudetto, but he revealed he is also enjoying life under new boss Simone Inzaghi.
"He's helping me so much," he told Inter's official website when asked about Inzaghi.
"We met when I came back from the Copa America and before that point, we'd already spoken quite a lot on the phone. He's helped me so much with my performances.
"I always try to learn as much as I can from coaches because they have a wealth of experience and they always try to do the best for their players."