Giles said after Sunday's 84-run T20 loss that he wanted the responsibility of coaching England in all three forms.
"I'd be very interested in doing the job, definitely. I'm not going to deny that and I'm sure I'll be applying for the post,'' Giles said.
"But before then, there's a lot to think about ... we've had a pretty big fall so we've got some turning around to do before Bangladesh (World T20 Championship), otherwise this sort of thing could happen again.''
Arthur had a successful tenure in charge of South Africa before taking over the Australian team in late 2011.
However, appointing Arthur would be a gamble after his time in charge of Australia ended in disaster last year.
After overseeing Australia's 4-0 whitewash at the hands of India, which included the homeworkgate fiasco, Arthur then led a failed campaign at the Champions Trophy.
Then just over two weeks before the Ashes were due to start, he was sacked and replaced by Darren Lehmann.
Cricket Australia (CA) suggested Arthur hadn't been strong enough to stop a negative culture gripping the national team.
Arthur left on good terms, before things turned bitter when he took Cricket Australia to the Fair Work Commission, demanding $4 million in compensation for his sacking.
CA and Arthur ultimately came to a settlement.
Flower stood down as England coach on Saturday, and could move into a development role with the ECB.
- AAP