US-backed Guatemala won all six rounds of balloting held on Tuesday but each time failed to secure the required two-thirds majority needed in the 192-member assembly to win one of the 10 rotating council seats for Latin America.
Voting in the assembly was set to resume on Wednesday afternoon a few hours after Guatemalan Foreign Minister Gert Rosenthal and his Venezuelan counterpart Nicolas Maduro were due to meet to try to find a compromise solution.
The contest to succeed Argentina on the council is now in its third week, despite intense lobbying by the United States, which backs Guatemala over Venezuela, whose president, Hugo Chavez, has a running battle of words with US President George W. Bush.
Venezuela has refused to drop out despite trailing Guatemala in all but one of the voting rounds.
Chile's UN Ambassador Heraldo Munoz told reporters that the time had come for an alternative solution.
He expressed hope that a new meeting between Rosenthal and Maduro could lead to "a consensus which is absolutely fundamental because as we have seen in the voting today, the votes are frozen, the shifts are minimal."
Munoz insisted that neither Guatemala nor Venezuela would be able to secure a two-thirds majority of the General Assembly.
"We have to think of the political alternative. A political solution can only be decided by the two countries involved and by their foreign ministers with the backing" of the Group of Latin American and Caribbean Countries (GRULAC), he added.
US Ambassador John Bolton restated US support for Guatemala.
"The traditional US posture is we don't get involved in regional group deliberations over Security Council candidacies," he noted. "As long as Guatemala is in the race we support Guatemala ... and we've expressed our concerns about Venezuela."
Venezuela has put out feelers to Bolivian President Evo Morales and the Dominican Republic's President Leonel Fernandez about their possible candidacies.

Copyright AFP (Agence France-Presse), 2006