The government claims that, during last week's meeting, Iceland Foods refused to give up control over the word and presented unsatisfactory proposals to resolve the situation.
As a result, the government will proceed with legal action to invalidate the trademark. It described the issue as a "matter of principle".
"The registration of a country name that enjoys highly positive national branding to a private company defies logic and is untenable as it hinders companies and entities to register their products with their country of origin," the government said in a press release.
Iceland Foods confirmed that it sent a delegation to Iceland on Friday to meet with the government, in the hope of achieving an amicable solution.
"This got nowhere because it rapidly became clear that the Icelandic authorities have no interest in reaching a compromise," said Iceland Founder & CEO Malcolm Walker, in a press release.
"We have no real idea why this has suddenly become such a major problem for Iceland (the country). Iceland Foods had Icelandic majority shareholders and Icelandic representatives on its board for seven years to 2012. At no point in all those years did any representative of Iceland (the country) raise the slightest concern about our company's branding."
Walker denied claims that the company has tried to prevent the country using the name "Iceland" to promote Iceland. He added that his company has only ever sought to prevent other food and retail companies using "Iceland" in a way that may cause confusion with its brand.
"The one absolute certainty about legal action is that it will take up a great deal of time and cost a large amount of money, and that ultimately the only real beneficiaries will be lawyers," he added.
"That is why I once again urge the Government and people of Iceland to talk to us to achieve the sensible coexistence agreement which I am sure is well within reach and very much in the best interests of all parties and their stakeholders."
Follow CNBC International on Twitter and Facebook.