The former First Minister announced a judge-led public inquiry on June 5, 2014, and later told MSPs it would be "swift and thorough".
It has since cost the public purse more than £12 million and has yet to report back.
He said: "The slow progress of this inquiry is just hugely frustrating for everybody that's involved.
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"It's cost the council, and I'm sure a whole number of other bodies, an awful lot of money, including the government spending millions of pounds on this inquiry."
Mr Day said the council is having to use public money to keep its legal team on standby, and insisted “the city and the public deserve better”.
He said: "I'll be writing to the government to say can the government, who called for the inquiry, make something happen.
"I appreciate it's an independent inquiry but there needs to be surely some timelines put on that, or else it's a never-ending process and a never-ending cost to the public purse, and I'm sure the public want to see an end to this as much as we do."
He added: "I think it's time for some action."
The tram inquiry, led by former Lord Advocate Lord Hardie, 76, has taken longer to conclude than the Chilcot inquiry into the Iraq war.
Scottish Conservative Lothians MSP Miles Briggs said: “It is simply astonishing and disgraceful that this inquiry has yet to report back.
"It has saddled taxpayers with a bill for more than £12 million so far and we still do not know it’s conclusions.
“Despite being announced to much fanfare, the inquiry has lurched from one disaster to another.
“Residents and businesses who have suffered continued disruption deserve this inquiry to finally see the light of day and give them answers at long last as to what went wrong with such a major project.
“At this rate we will soon hear calls for an inquiry into the inquiry.”
A spokesman for the inquiry said: “Lord Hardie’s remit is to conduct a robust inquiry and he has made it clear it will take as long as is necessary to get the answers the public wants in relation to the issues surrounding the Edinburgh trams project.
"The inquiry’s evidential database contains over three million documents that have to be carefully considered which is an extensive but vital task.
“We continue to make good progress towards producing the report and recommendations which will be published as soon as practicable.
“The Inquiries Act 2005 obliges the chair to consider cost at all times since it is funded from the public purse.
"Throughout the process all efforts have been made to ensure the inquiry delivers best value.”
A Transport Scotland spokesman said it “agreed to continue to fund the costs of the Edinburgh trams inquiry until it is completed”.
He said: “Staffing matters, including how many people the inquiry employs and how much they are paid, are a matter for the inquiry. This is because, as a statutory inquiry, it is independent of the Scottish Government.”