Marketing Edinburgh, the council-funded firm in charge of the strategy, is re-thinking its approach amid claims the proposed slogan is "absolutely appalling" and "twee".
Other catchphrases, such as "Paintthetownredinburgh" and "Wellfedinburgh" – for the city's pub and club scene, and its restaurants – have also been floated but met with little enthusiasm.
The row means the city's long-awaited re-branding has been scaled back, with a more limited winter campaign being planned instead.
Edinburgh's business and political communities have long hankered after a catchy logo to help transform the city's image. The "Glasgow's Miles Better" campaign, launched in 1983, is considered the standard-bearer for a Scottish city. It helped soften Glasgow's hard-boiled image by focusing on its cultural treasures and won international awards.
The "I Love New York" campaign was another global success.
Marketing Edinburgh was set up last year with council money to promote Edinburgh as a destination. Its chief executive is Lucy Bird; city council chief executive Sue Bruce is a director.
The Sunday Herald understands Marketing Edinburgh has a six-figure sum at its disposal to produce a winning slogan and strategy for the capital. The Leith Agency, which counts the Scottish Government, Honda and Grolsch among its many clients, is the firm that Bird and her colleagues have been using.
At a presentation to senior councillors earlier this year, the arms-length body unveiled "Incredinburgh" as the preferred option.
Other sub-slogans in a similar vein have also been suggested. 'Romanceisnotdedinburgh' was one of those mooted, as was 'Grabyoursledinburgh' for winter sports.
The presentation was met with scepticism by the politicians, who asked for other ideas to be considered. A second meeting with councillors led instead to the same theme being re-heated, prompting deputy council leader Steve Cardownie to walk out.
The new strategy is now said to be in limbo, despite the fact that Marketing Edinburgh is planning to launch a campaign in two weeks. Rather than unveil a year-long advertising campaign, Marketing Edinburgh will instead launch a winter-only strategy.
A council source said: "The politicians did not think the proposed campaign lived up to expectations, and it caused a huge amount of friction. The council insisted the campaign should be for the winter only, with the other slogans being put out for market testing."
Although no final decision has been made about the slogans, the Leith Agency has left a couple of digital clues about the plan.
On July 2 this year, the Edinburgh agency registered the web domain name Incredinburgh.com. Ten days later, a Twitter post in the name of Leith Agency creative director Gerry Farrell described a capital landscape as "Incredinburgh".
Jenny Dawe, the former Liberal Democrat leader of the council, said: "I think these ideas are absolutely appalling. You don't need silly slogans to market Edinburgh. They don't sound worth using at all, and they make me shudder."
Cameron Rose, the Conservative group leader, said: "Some of the slogans are a bit twee, and when you first hear 'Incredinburgh', I'm afraid it just doesn't cut it. That said, some may grow on us."
Steve Burgess, convener of the council's Scottish Green group, said: "Our group has never been convinced about the amount spent on marketing, as Edinburgh as a city speaks for itself."
Marketing Edinburgh's Lucy Bird said: "Over the past few months, we've collaborated with many of our city partners to develop a dynamic and lively new campaign for Edinburgh. As you would expect, there's a huge amount of planning and discussion around a project like this and we're welcoming feedback."
A council spokesperson said: "As with all marketing campaigns, ideas evolve during the discussions between partner organisations. We - look forward to the launch of the new campaign this winter."