The tourists ended their disastrous tour of Australia with a crushing 84-run loss in the third Twenty20 international in Sydney on Sunday night.
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They also lost the Ashes 5-0 and the ODI series 4-1 to go along with their 3-0 defeat in the Twenty20s.
Giles, 40, is the incumbent England Twenty20 and ODI coach and admits he was surprised by Flower's decision to resign.
Under the new structure, Flower's replacement will be responsible for overseeing all three formats including the Test team.
"I'd be very interested in doing the job, definitely," Giles said after England
"I'm not going to deny that and I'm sure I'll be applying for the post.
"It's one of the biggest jobs in cricket and I'd apply for it because I want to do it.
"As someone who was very proud to play for England, I've been proud to coach the one-day side and to do the whole job would be a fantastic opportunity."
The former off-spinner added of England's woeful showing in all three formats in Australia: "We've come from a position where we won the Ashes last year, were runners-up in the Champions Trophy and a confident T20 side, and we've had a big fall during this winter.
"So we've got some turning around to do and that starts in the West Indies and obviously in the immediate term my role is to lead that side to the West Indies and the World Cup in Bangladesh.
"And we've got a lot of hard work to do before we get to Bangladesh, otherwise this sort of thing is going to happen again."
The task of finding Flower's replacement will begin this week.
Giles insists his focus is on preparing for the upcoming limited-overs tour to the West Indies and the World T20 starting in Bangladesh on March 16.
"I think it is important that I control what is coming up in the next couple of months," he said.
"If I have to put an application in over the next couple of weeks then I'll think about that and get it done.
"Results will talk as well and I've got a really big series in the West Indies and more importantly the World Cup in Bangladesh.
"Our record in the sub-continent for limited-overs cricket isn't brilliant so it is a good challenge.
"We're pretty much at base level right now. We have a lot of work to do and before that we have a lot of talking to do about which squad we take."