Prior was dropped for Bairstow for the fourth Test in Melbourne after an indifferent series but his understudy gave a mixed performance, dropping a catch and failing to respond to a snick from Chris Rogers when he was on 19, with the opener going on to score a century. (Agencies)
Clarke said Prior's vast experience was needed by England as they seek to deny Australia a 5-0 whitewash in Sydney, but Cook said Bairstow would get another chance as wicketkeeper.
"I think Jonny deserves another chance. It's very difficult to come in for one Test match," Cook said on Thursday. "We know he's a very talented cricketer, he's got another opportunity in this Test match to show that."
Australian captain Clarke said England missed the big-match experience of Prior, who has 75 Test caps.
"I would be surprised if Matt Prior doesn't get recalled back into their team. Being a senior player with the amount of experience he has, I think they've certainly missed that," Clarke said at his media conference on the eve of the Test.
"He's been an important player for England over a long period of time. I think you can see the benefit of (wicketkeeper) Brad Haddin in our team, just with his experience," he added.
But while Prior has an impressive Test batting average of more than 40, Cook said he did not deserve to be in the England team on form.
"He's still a fantastic cricketer, he's 31. He's still got a lot of Test cricket available to him if he wants it," the England captain said.
There is speculation that England could field fresh faces in Sydney, with Gary Ballance, Scott Borthwick and Boyd Rankin -- all yet to play Test cricket -- in the frame. Cook did not discount the possibility.
"I think it's possible, yes. We need to assess (spinner) Monty (Panesar) after training today. He's had a bit of a sore calf from yesterday but we will see how he pulls through training today and make a decision on him either today or tomorrow," he said.
Cook, who has faced criticism of his captaincy, particularly from Australian spin legend Shane Warne, said he had to be true to himself.
"I have to be the man I have to be. I can't change because Shane Warne says I need to change totally. I can certainly look at his stuff and he might have a point on some of it," he said.
"But if I listen to every person, your whole mind gets muddled anyway. I have to be true to myself, think about what's best for me and this England side.”
"That nature of a captain especially is there's a number of decisions where you could go a different way. That's why you're responsible for making those decisions. You can't be thinking someone in the commentary box might be thinking something else."
Warne has often criticized Cook as a negative skipper with his field placements and says he finds his tactics boring and unimaginative.
Prior was dropped for Bairstow for the fourth Test in Melbourne after an indifferent series but his understudy gave a mixed performance, dropping a catch and failing to respond to a snick from Chris Rogers when he was on 19, with the opener going on to score a century.