Mr Trump's campaign planned to announce his selection on Friday, but cancelled the event because of the attack in Nice, France.
Sources close to campaign told ABC News that Mr Pence accepted the offer.
However, Mr Trump told Fox News on Thursday evening: "I haven't made my final, final decision."
Mr Trump hopes Mr Pence can help him appeal to the party's conservatives.
Other candidates reportedly in the running were former House Speaker Newt Gingrich and New Jersey Governor Chris Christie.