Kirk's killer has eluded police and federal agents for more than 24 hours after Wednesday's shooting, in which a sniper fired a single gunshot that killed Kirk, 31, during an appearance at Utah Valley University in Orem.
"I think we have him," Trump told Fox News in an interview on Friday, adding that a person who knows the suspect had turned him in.
"I think with a high degree of certainty we have him in custody."
Trump did not share the suspect's identity.
Previously, US investigators said they had found the bolt-action rifle believed to have been used to kill Kirk and released images of a person of interest.
The FBI had circulated grainy images apparently taken from security cameras showing a person wearing a black top, black sunglasses and a dark baseball cap.
The long-sleeved top appears to have been emblazoned with an image of a bald eagle flying across a US flag.
Kirk, an author, podcast host and close ally of Trump, helped build the Republican Party's support among younger voters.
Trump said the suspect was recognised by people close to him, and suggested the suspect's father drove him to a police station and handed him over.
FBI and state officials said the killer arrived on the campus a few minutes before the start of the event, a debate led by Kirk titled "Prove Me Wrong" in front of 3000 people at Utah Valley, south of Salt Lake City.
Security-camera videos showed a person going up stairwells to gain access to a roof before firing at Kirk, the officials said at a media briefing.
Kirk, a staunch defender of gun rights, was answering an audience question about mass shootings when the bullet struck his neck.
Audience members fled in panic.
The shooter jumped off the roof and fled into an adjoining neighbourhood, said Robert Bohls, an FBI special agent in charge.
Investigators found a "high-powered, bolt-action" rifle in a nearby wooded area, and were examining that along with palm prints and footprints for clues.
On Thursday, with classes cancelled, the roof of the building on the otherwise deserted campus and the nearby woods were strung with yellow tape as investigators scoured them for evidence.
The shooter appeared to be of college age and "blended in well" on the campus, Utah Public Safety Commissioner Beau Mason said at the briefing.
Kirk was the co-founder and president of the conservative student group Turning Point USA and appeared at Utah Valley on Wednesday as part of a planned 15-event "American Comeback Tour" of US college campuses.
His killing stirred outrage and denunciations of political violence from Democrats, Republicans and foreign governments.
Trump said he would award Kirk the Presidential Medal of Freedom, the nation's highest civilian honour.
Kirk, who was married and the father of two young children, was celebrated by Republicans as a charismatic advocate for right-wing policies on race, gender, immigration, religion and gun regulation.
He frequently engaged with his critics from the far left to the far right, often inviting members of his audiences to debate him live.
