Charlie Wood and Ed Bartlam say there is a pressing need for the first “concerted and co-ordinated” campaign to promote all of the festivals under the one banner properly.
But they are adamant it should not be brought in if it means public funding for the festivals is cut back.
The city council has been lobbying the Scottish Government for the powers to bring in a “transient visitor levy” for more than a decade.
However, the idea has been consistently opposed by ministers and the main Scottish tourism bodies.
Despite the lack of support, council leader Adam McVey has embarked on a new charm offensive to secure new support for the initiative, which could see £1 or £2 a night added to the cost of hotel rooms across the city.
Mr Wood said: “
“It doesn’t put tourists off going to Paris, Rome or Venice. Why is going to put people off coming to Edinburgh when there is more to do than any of these cities? It’s something we can only advocate for. Clearly it’s a decision that has to be made by the Scottish Government, but we’re a big supporter of the idea.
“We have an interest in the Fringe and the winter festivals and are part of Festivals Edinburgh. Anything that supports the whole festival ecology in Edinburgh has to be a good thing, but the one thing it shouldn’t do is replace any existing funding.”
Mr Bartlam said: “A tourist tax shouldn’t be seen as a replacement for funding that is currently there, but it’s a simple fact that more funding is needed for wider cross-city initiatives to support all of the festivals.
“There is a lot more that can be done on joint marketing across all the festivals.
The Underbelly founders said money raised from an Edinburgh tourist tax could help pay for infrastructure improvements for the festivals, including a long-awaited joint box office system.
However, William Burdett-Coutts, artistic director of the Assembly Rooms, said: “In principle, a tourist tax is a great idea, but in practice I’m not sure about adding even more cost to the price of hotel rooms in Edinburgh, which are already exorbitant as it is, and then there is the issue of how the money raised is disseminated.
“The Fringe Society is seen as the organisations which holds the whole thing together and tends to get the funding, but the work is actually done by the venues.
“I would believe in a tourist tax if some of the money actually went into shows that would keep the festival lively and encourage work that would not normally come.”