More than 7,000 leads and tips had poured in, officials said. Authorities have yet to publicly name the suspect or cite a motive in the killing, the latest act of political violence to convulse the United States.
"You have to go forward," the president said when asked about appearances getting cancelled after Kirk's assassination.
State and federal officials in Utah are planning to hold a news conference at 9 a.m. ET on the investigation.
Utah Gov. Spencer Cox and Public Safety Commissioner Beau Mason, FBI Director Kash Patel and FBI Special Agent in Charge Robert Bohls are expected to attend.
"I didn't want to remember Charlie that way," Trump said in an interview on Fox News Channel's "Fox & Friends."
Graphic video of the shooting in Utah has circulated widely online.