The 37-year-old Argentine star went down in the 64th minute of a 1-0 victory over Colombia, hobbling off the field with cameras showing a huge swollen ankle.
Messi is set to miss Inter Miami home matches on Wednesday against Toronto FC and on Saturday against the Chicago Fire.
"There will be further testing to determine the severity of the situation," Martino said. "He has to have more tests and wait for the results."
Messi, in an Instagram posting, was thankful for messages of concern and support he received after Sunday's match.
"I want is to thank everyone for the messages and greetings. I'm fine, thank God, and I hope I can soon be on the field again enjoying what I like to do most," Messi wrote.
With 14 wins and five draws from 23 matches, Inter Miami ranks second in MLS and the Eastern Conference on 47 points, one behind Cincinnati, which routed Miami 6-1 on July 6.
Inter Miami is set to begin defending the Leagues Cup title against MLS and Mexican league rivals next week.
