The council lead said First Minister Nicola Sturgeon has confirmed that ministers will lay a bill before parliament early next year.
But, writing in the Edinburgh Evening News he said he is “determined” to see the tax implemented as soon as possible.
Mr Day “estimates that a levy could raise in the region of £15million per year” – funding he believes “could secure additional resources to invest sustainably in and managing the success of tourism”.
He added: “I want to be clear that these any visitor levy would be for the benefit of all of Edinburgh, not just the city centre.
“All the research suggests a small levy would not deter tourists from visiting the Capital.”
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