Havana Grey is a winner of four of his six races to date and landed the Molecomb Stakes at Goodwood recently.
Unfortunately, a son of Society Rock, took his form to a new level when winning the Prix Robert Papin at Maisons-Laffitte last month.
"I would like to split them up, but this race looks the right one for both of them at this stage of their careers," said Burke.
"If the ground was to turn up quick, which is unlikely at this stage, then Unfortunately is actually in the Gimcrack at York, so we could wait for that with him.
"I can't see a reason why Havana Grey won't stay six furlongs. Before Goodwood there might have been a doubt, but the ground was very testing that day and he powered through the line.
"Also the Morny is 1200 metres, which isn't quite six furlongs.
"It's the logical place to run them both and I've been happy with the pair of them since their last runs."