The event has already been pushed back two weeks, although Rio organising committee president Carlos Nuzman said in London this week that the event would go ahead on April 29-May 1.
Speaking at the world track championships on Saturday, Cookson said he thought that date would be "a challenge".
However, he said he believed Rio would be ready.
"I have no doubt there will be a velodrome ready and functional to the required standards for the Olympic Games," he said. "There is no alternative. That's the straight answer."