The Wallabies fly to Europe for their five-match tour on Friday and a squad had been scheduled to be selected by Tuesday.
"In the ideal world we will identify a coach who will take us through to the World Cup and beyond, in the absence of that we will look for an interim measure but I hope it doesn't come to that," Pulver said.
Springboks World Cup-winning coach Jake White, who lost out to McKenzie as Robbie Deans' replacement last year, remains an option after being released by the Durban-based Sharks.
But White had been overlooked largely due to his conservative approach, while the popular Cheika is renowned for instilling a successful attacking approach that took NSW to a breakthrough Super Rugby crown.
Although Cheika is contracted to the Waratahs for next year, it's understood he would answer any national call.
He's privately known to covet the position but taking over from a former Randwick teammate in such dramatic circumstances is not how he'd like to accept the job.
Cheika had been hoping to fulfil his NSW commitments and viewed the Wallabies position as a job further down the track.
The only coach to win a Super Rugby title and the European Cup, he told AAP earlier this week he hadn't spoken to anyone at the ARU about any national role.
Board member Eales had been influential in preferring McKenzie over the South African White.
On Saturday night, the former Wallabies skipper said Australian rugby didn't need the most convenient decision, it needed the best decision.
"And it's tough to be able to do that in the right time," Eales told Fox Sports. "There's a lot of people looking at it and we'll look at it from a very thorough perspective."
-AAP
