Kirk, a prominent conservative activist with close ties to Trump, was assassinated on September 10 while speaking on a college campus in Utah.
A 22-year-old technical college student has been charged with Kirk's murder.
Investigators are still looking for a motive and have not said the suspect operated in concert with any groups.
But the Trump administration has used the killing as a pretext to revive years-old plans to target left-wing groups they regard as being hostile to conservative views.
Antifa, short for anti-fascist, is a "decentralised, leaderless movement composed of loose collections of groups, networks and individuals", according to the Anti-Defamation League, which tracks extremists.
"While some extreme actors who claim to be affiliated with antifa do engage in violence or vandalism at rallies and events, this is not the norm," it says on its website.
Trump's 370-word executive order on Monday directs "all relevant executive departments and agencies" to "investigate, disrupt, and dismantle any and all illegal operations" conducted by antifa or anyone who funds such actions, according to the White House.
"Individuals associated with and acting on behalf of Antifa further co-ordinate with other organisations and entities for the purpose of spreading, fomenting, and advancing political violence and suppressing lawful political speech."
Federal law enforcement officials already investigate violent and organised crime associated with a variety of hate groups and ideological movements.
The US government does not officially designate solely domestic groups as terrorist organisations in large part because of constitutional protections.
But a Justice Department official with knowledge of discussions on the issue said Trump's order would unlock expansive investigative and surveillance authorities and powers.
The person, who declined to be named, said the designation would allow the US government to more closely track the finances and movements of US citizens and to investigate any foreign ties of the loose network of groups and non-profits the Trump administration views as antifa.
It was not clear which individuals would be the target of such a probe.
Critics of the administration have warned it might pursue an attack on free speech and opponents of the Republican president.
Political violence experts and US law enforcement officials have previously identified far-right attacks as the leading source of domestic violent extremism.
Trump administration officials have sought to portray left-wing groups as the main drivers of political violence in their remarks since Kirk's death.
Legal experts have said the domestic terrorism designation might be legally and constitutionally dubious, hard to execute and raise free-speech concerns, given that subscription to an ideology is not generally considered criminal under US law.
During the first Trump administration there were at least two failed efforts to designate antifa a terrorist organisation.
