Martino was announced as the new boss of Miami on Wednesday, where he is set for a reunion with Messi and Busquets, whom he previously coached at Barcelona.
"When we spoke with Leo, yesterday I spoke with Sergio, we talked about arriving to have success, to compete and compete well," Martino said during his introductory news conference on Thursday.
"This happens to us even on a personal level, sometimes we associate the United States, Miami, holidays --- this isn't that. We want to compete... they are not footballers who are going to come here to not compete."
Earlier this month, Messi announced his decision to join Miami, which was then followed by last week's news that midfielder Busquets would also join the MLS club. Martino coached both players during his managerial stint in Barcelona from 2013 to 2014, as well as Messi at the international level with Argentina from 2014 to '16.
However, Martino, who returns to MLS after leading Atlanta United in the 2017 and 2018 seasons, said discussions with Miami began before the deals for both players.
"I started the talks before [the additions of Messi and Busquets]," said the Argentine. "I always found this opportunity attractive...let's just say that the presents arrived next."
With the arrival of the global superstar and 2022 World Cup champion to MLS, Martino said Messi, 36, could help further inspire the growth of soccer in the United States.
"I've read that NFL and NBA players are talking about this situation, which seems very rare," he said. "I believe that MLS has a constant evolution and the examples are the great soccer players from the United States who are playing in Europe, and in very good teams in Europe. There is already a process that has started.
"[And] that the best player in the world decides to play in this league, obviously it will open up an even better panorama for that growth. I believe that the United States is not a country that misses out on this type of opportunity."
At the MLS level, Martino and his new players will need to help revive a team that sits in last place in the Eastern Conference and is on a seven-game losing streak. Miami has 16 games left in the regular season and needs to move from 15th in the table to a spot in the top nine to qualify for the 2023 MLS Cup playoffs.
Chris Henderson, chief soccer officer and sporting director for Miami, stated Thursday that they're working on the visa process for Martino, who could be ready to work during "the next five, six days."
As for Messi's debut, Henderson added that it could take at least a couple of weeks. Miami managing owner Jorge Mas revealed last week that they're expecting Messi to play his first game against Liga MX's Cruz Azul in a Leagues Cup match on July 21.
"With regard to Leo, terms are agreed, but we're working on the paperwork with Major League Soccer, so those are going to take time to finish, but we hope that there's a time in mid, late July that he's ready to go," Henderson said.
Martino has replaced English manager Phil Neville, who was fired by Miami earlier this month after leading the team since 2021.
Before taking charge of Miami, Martino coached Mexico's national team from 2019 through the end of the 2022 World Cup. Following a disappointing exit in the group stage in Qatar, Martino left his role at the end of his contract.