Soldiers from Venezuela's national guard have deserted their posts ahead of an opposition-led effort to bring aid into the economically-devastated county.
The soldiers have asked for help from neighbouring Colombia's migration agency, the department said.
Separately Venezuelan troops have fired tear gas at people attempting to cross into Colombia to work.
President Nicolas Maduro has ordered the Venezuelan military to close the border with Colombia to bar humanitarian aid, which is being supported by the opposition, from entering the country.
Venezuelan forces on Saturday hurled tear gas and fired rubber to break up a crowd demanding to cross the Urena border bridge to Colombia.
"We want to work!" people chanted as they faced Venezuelan National Guard riot police blocking the crossing, one of several ordered closed by Mr Maduro late on Friday.
Opposition leader Juan Guaido, who has declared himself interim president, is attempting to cross into Venezuela from the Colombian border on Saturday with thousands of volunteers carrying emergency supplies.
While the need for basic food and medicines is real, the effort is also meant to embarrass military officers who continue to support Mr Maduro's increasingly isolated government.
Mr Guaido, recognised by most Western nations as the country's legitimate head of state, defied court orders not to leave Venezuela by arriving on Friday in the Colombian border city of Cucuta, where aid from the US and Colombian governments is stockpiled in warehouses.
The opposition leader invoked articles of the constitution in January to assume interim presidency and denounced Mr Maduro as a usurper, arguing his 2018 re-election was illegitimate.
President Maduro blames the country's dire situation on US sanctions that have blocked the country from obtaining financing and have hobbled the OPEC nation's oil industry.
Violence has already flared in some border regions. Police firedIn Two people were killed on Friday night when the Venezuelan army opened fire in an village near the Brazilian border after indigenous leaders attempted to prevent them from advancing, killing a woman and her husband.
Nearly 200,000 people attended a benefit concert in Cucuta on Friday featuring Latin pop stars, including Luis Fonsi of "Despacito" fame, many of whom called on Mr Maduro to step down.
A rival concert held by the ruling Socialist Party on the Venezuelan side was sparsely attended.
Brazil's humanitarian aid has arrived on the border
The first truck with humanitarian aid from the Brazilian government has arrived in the city of Pacaraima on the border with Venezuela.
The crossing has been closed on orders from Venezuelan President Nicolas Maduro and the truck loaded with food and medicine will now wait in Brazilian territory.
Brazil's Foreign Minister Ernesto Araujo said he expects Mr Maduro's government to allow the aid to pass.
"It is very exciting to see people anxious to recover their freedom and have a decent life," Mr Araujo said.
Venezuelan opposition leader arrives at the border
The Venezuelan opposition leader has arrived at the border with the leaders of Colombia, Chile and Paraguay.
Defected soldiers named
We now have the names of three of the four national guard members who defected earlier this morning, Harriet Alexander reports from the border.
Three drove two tanks across the Simon Bolivar bridge, shunting the barricades out of the way to hand themselves in. The fourth walked across the Francisco de Paula Santander bridge, to huge applause.
The three have been named as Lt Richard Sánchez Zambrano, Sgt Major Edgar Torres Valera and Sgt Major Oscar Suárez Torres.