The club raised the matter following the announcement that the former Hearts and St Mirren striker had joined the West Lothian side on a two-year deal, with the option of a third, after arriving from Bangladeshi outfit Sheikh Russel.
The 30-year-old Portuguese forward left Scottish football in January 2018 amid claims he had suffered racist abuse at the hands of "a minority" of Hearts fans, prompting an apology from the Tynecastle club.
A Livingston statement read: "The club is aware of a tweet containing racial abuse aimed at new signing Esmaël Gonçalves.
"We’ll be following the relevant protocol to ensure this is dealt with appropriately and pushing to ensure the Twitter user is punished accordingly."
Goncalves helped St Mirren win the League Cup in 2013, scoring eight goals in 16 games, while he netted 15 in 42 for Hearts.
Livi boss David Martindale is hopeful he can find that form in his attack next season.
“I’m really happy to get this one over the line," he told the Livingston website.
“Esma can play anywhere across the front 3 but was more predominantly used as a number 9 in his previous spells in Scotland.
“He knows the league well and likes the physicality the Scottish league will throw at him and has an eye for a goal.
“I’m confident that Esma, once up-to-speed, will prove to be a valuable member of the first-team squad.”