Current boss Matthew Mott is under pressure amid a review of England's disappointing performances in their last two major tournaments.
England fell at the group stage at the 2023 World Cup and were unconvincing in reaching the semi-finals of the T20 equivalent, before losing heavily to India last month.
Morgan, who captained England to their World Cup triumph in 2019, is considered a potential replacement for Mott but he has indicated he is focused on his media and family commitments.
Morgan, working as a pundit during TV coverage of The Hundred in London on Tuesday, said: "I've been asked a lot over the past couple of months about the role and if I would take it on, and my answer has simply been that the timing for everything in my life at the moment is not right.
"Yes, I want to coach down the line, but I have a young family and I spend a lot more time at home and watching cricket doing this. I'm absolutely loving what I'm doing.
"This news is actually news to me. It's obviously not nice when a coach comes under fire and there is a lot of speculation about his future but only time will tell what will happen."
England's white-ball side are next in action when they face Australia in September, with their next major tournament being the ICC Champions Trophy in February 2025.
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