Several tons of medicine, food and basic items have been stocked since Thursday in Cucuta, close to the border crossing point of Tienditas which Venezuelan soldiers have blocked with freight containers.
The US has presented a draft resolution at the United Nations Security Council calling for global aid deliveries and a presidential vote in Venezuela, triggering a Russian counter-proposal.
The military's backing is critical to the sway of power in Venezuela.
The large-scale military exercise, which is being held in the country's 200-year history, will continue until February 15.
Shipments of US food, medicine, and humanitarian aid are on the Colombian side, with a tanker and two large trucks sitting in the middle of the bridge.
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UN Secretary General Antonio Guterres has also been urged by the U.S. to "utilize his good offices" to obtain such elections, stressing "the need to prevent further deterioration of the humanitarian situation in Venezuela and to facilitate access and delivery of assistance to all in need in the entirety of the territory of Venezuela".
Suffering the worst crisis of its modern history, Venezuelans have had to grapple with life-threatening scarcities amid eye-popping levels of hyperinflation that have rendered salaries and savings worthless.
The United States and dozens of other countries have recognized Guaido as the nation's legitimate head of state.
It also "calls for the immediate start of a political process leading to free, fair and credible presidential elections, with worldwide electoral observation, in line with Venezuela's constitution".
Tensions rose when opposition leader Guaido declared himself acting president on January 23, a move which was supported by the US and many European and Latin American countries.