Previously, the briefing was set to start at 12:45 local time (19:45 GMT).
"Due to the rapid developments in our investigation, we will suspend this. We will release an updated time later today," the notice read, without clarifying the nature of the developments.
Earlier in the day, the FBI announced a reward of up to $100,000 for information that will lead to the identification or arrest of a person or individuals responsible for killing Kirk.
Kirk was assassinated at a university in Orem, Utah on Wednesday while giving a speech.
No suspects were in custody as of Thursday morning, but an image bearing a 'person of interest' has been released.