Growing numbers are fleeing economic meltdown and political turmoil in the South American nation, threatening to overwhelm their neighbouring countries.
Colombian officials will hold a meeting in the country's capital Bogota with their counterparts from Ecuador and Peru to discuss the issue.
Ecuador and Peru have this month tightened entry rules for Venezuelans, requiring them to carry passports instead of just national ID cards. In Brazil, rioters drove hundreds back over the border.
Describing those events as early warning signs, International Organisation for Migration (IOM) spokesman Joel Millman said funding and means of managing the outflow must be mobilised.
“This is building to a crisis moment that we’ve seen in other parts of the world, particularly in the Mediterranean,” he said.
The IOM and UN refugee agency UNHCR also called on Latin American countries to ease entry for Venezuelans, more than 1.6 million of whom have left since 2015.
Peru’s top immigration official, Eduardo Sevilla, said his country would exempt some Venezuelans from the passport requirement, including parents with children seeking to join the rest of their family, pregnant women and the gravely ill.
But Mr Sevilla said authorities would also be vigilant of attempts to evade the new rule by claiming refugee status.
“Is UNHCR going to take responsibility if that person commits a crime,” he said. “Our priority is to contribute to security and internal order by clearly identifying people.”
UNHCR spokesman Andrej Mahecic said that governments had made “commendable” efforts despite some reception capacities and services being overwhelmed. But he added that “some disturbing images” had emerged from the region in the past week which risked stigmatising Venezuelans who had fled and complicating efforts to integrate them.
Their comments came as thousands of Venezuelans rushed to reach Peru before the new rules came into effect.
Tired men, women and children, many lugging their belongings in suitcases, lined up at the Peru-Ecuador border to pass through migration authorities.
Peru's economy is expected to grow by four per cent this year and many Venezuelans already have at least one relative living in the nation. Some 73,000 Venezuelans have been given a temporary legal status allowing them to live and work in the country while another 108,000 have applied.
Reuters contributed to this report
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