Inter Miami managing owner Jorge Mas says the club will announce Martino's successor "in the next few days", with another Argentine, Javier Mascherano, heavily linked.
The former Barcelona and Liverpool midfielder is a former team-mate of both Inter Miami captain Lionel Messi and striker Luis Suarez.
"I'm not going to discuss names," added Mas. "We are very near [the] end of process, we will be announcing a new coach in the coming days.
"We got it down to three final candidates by Tuesday. We made a decision who we wanted the day before yesterday and we will announce in the next few days."
Inter Miami were bottom of the MLS standings, having lost 13 of their 18 fixtures, when Martino was appointed as Phil Neville's successor in July, 2023.
He quickly improved their fortunes by sealing the Leagues Cup and further built on that success in 2024, with Inter Miami setting the record for the most single-season MLS points en route to winning the Supporters' Shield.
However, they subsequently came up short in this season's post-season play-offs, losing against Atlanta United in the first round.
Mas said Messi had "given his input" into the process and he was keen for the Argentine forward to be "comfortable" with the new appointment, but had not been given power of veto.
Messi's contract is due to expire at the end of the 2025 campaign but Mas said he 'fully expected' the 37-year-old still to be with Inter Miami when the club moves into its new stadium in 2026.
The club has also controversially qualified for next year's Club World Cup after winning this season's MLS Supporters Shield for the team with the best regular season record.