Mesic's Office said in a statement on Tuesday that Crnic had declined the offer.
Crnic's term expires on May 15, but he will continue serving as a Supreme Court judge, a wish he expressed earlier.
Mesic will soon propose another candidate for the President of the Supreme Court in accordance with his constitutional powers, the statement said.
Under the Constitution, the President of the Supreme Court is elected by the Croatian Parliament at the proposal of the President of the Republic and with a prior opinion of the general session of the Supreme Court and the authorised parliamentary committee.