They said the Transient Visitor Levy will raise revenue from visitors to help maintain public spaces, transport links and the cultural sector.
Edinburgh is currently run by an SNP/Labour coalition.
The new charge would be £2 per room per night and would be capped at £14.
In 2019, the last year before the pandemic, Edinburgh welcomed 4.9 million visitors, who each stayed four or five nights on average.
A tourist tax was supported by 90 per cent of residents in a consultation held by Edinburgh Council in 2018.
Legislation to give councils the power to introduce a tourist tax was previously halted by the Covid pandemic, but work has since resumed.
Visitor charges are common across Europe. In Edinburgh, the tax would include holiday lets provided through companies such as Airbnb.
Only campsites would be exempt.
Adam McVey, leader of the SNP on Edinburgh Council, said: “Edinburgh is a world-class destination for tourists.
"The festivals are the largest of their kind anywhere in the world and, in most years, Edinburgh Castle is the most visited attraction in the UK outside London.
"The sector employs tens of thousands of people.
“But we all know that this footfall also strains the city’s capacity. The capital has to work hard to maintain the infrastructure that supports being a tourism hotspot – from keeping the streets clean to providing the transport links essential to getting around.
"And we also have to find ways to keep investing in our cities’ parks, museums, facilities like public toilets, and culture if we’re going to stay in the premier league of holiday destinations.
“While the Conservatives would just see city residents keep footing the bill, SNP councillors will introduce a fairer alternative.
“Anyone who has been on a city break in Europe knows that per-night charges to tourists are normal.
"Cities around the continent have realised that this is a simple, fair and accepted way of raising money to help deal with the pressure that comes with being a popular destination.”
He added: “The number one concern right now for business is staff availability, so some of this revenue could also be used to support people into these jobs, not only helping businesses but helping people across the city into fair work.”
Scottish Tory MSP Miles Briggs insisted any plans for a tourist tax "would be a serious blow to businesses across Edinburgh".
He said: “The SNP are woefully out of touch if they think introducing this will help the capital’s recovery in any way.
“Scottish Conservative councillors will remain firmly opposed to any harmful plans for a tourist tax brought forward by any SNP administration."
He added: “This is all too typical of the SNP’s failure to fully support and engage with businesses.
"Instead they would clearly to punish them with a further increase in their costs at a time when bills are already skyrocketing.”