The vehicles are currently parked near the Tienditas bridge, which remains blocked by Venezuelan troops.
President Nicolás Maduro, who has the support of the army, has rejected letting it into the country.
Opposition leader Juan Guaidó, who has declared himself interim president, has warned many Venezuelans are in danger of dying without international aid.
The bridge of desperation
Why the military is backing Maduro
Mr Guaidó is head of Venezuela's National Assembly and says the constitution allows him to assume power temporarily when the president is deemed illegitimate.
He has secured the backing of over 40 countries, including the US and most Latin American and European nations. Mr Maduro still has the support of China and Russia.
In other developments:
The US announced new sanctions on members of the Maduro administration, saying visas for members of the government-controlled Constituent Assembly would be revoked
European and Latin American members of the so-called International Contact Group said it was crucial to restore full democracy in Venezuela, saying they would push for new elections as soon as possible
What's the latest on the US humanitarian aid?
On Thursday, several lorries with food and medicine arrived at a collection centre in Cúcuta.
The vehicles were escorted by Colombian police motorcycles.
It was not immediately clear how the humanitarian aid would be delivered across the border.
The Venezuelan military had earlier placed cargo containers and a tanker lorry across the Tienditas bridge, which connects Cúcuta and the city of Ureña in Venezuela.
Mr Maduro has refused foreign aid supplies, saying they would open the way for US military intervention to oust him.
He said that "no-one will enter, not one invading soldier".
Meanwhile, US Secretary of State Mike Pompeo is demanding that Venezuela re-open the bridge, saying "the Maduro regime must let the aid reach the starving people".