The number of Venezuelan refugees represents around 12.5 percent of the national population and is double the size of the population of Caracas.
The director of the foreign ministry's Latin American department said Russian specialists would be sent to Venezuela if requested.
The president is reportedly frustrated with national security adviser John Bolton's overconfidence about toppling Venezuela's President Nicolás Maduro.
The Venezuelan opposition leader said he was focused on a "peaceful transition" but did not rule out foreign military intervention to oust President Nicolás Maduro.
"Tell the Bolivarian government and Nicolas Maduro what is necessary to be changed," the Venezuelan leader said.
Secretary of State Mike Pompeo suggested that President Nicolás Maduro was ready to flee the country following the attempted coup.
"Are we gonna be a hemisphere dominated by China, Russia and dictators?" GOP Representative Adam Kinzinger asked. "Or are we going to be a hemisphere dominated by freedom and democracy?"
"You have seen the Maduros around the world who just cling to their power, no matter how they treated their own people. And it's just time for him to go," said Kellyanne Conway.
Juan Guaidó, Venezuela's self-declared interim president, released a video from an air base near Caracas calling on Venezuelans to help topple President Nicolás Maduro.
Sources with knowledge of the plan say Blackwater's Erik Prince wants to send up to 5,000 Spanish-speaking troops into Venezuela.
Russian Deputy Foreign Minister Sergei Ryabkov said the new measures are illegal.
The group of 12 politicians asked Donald Trump to use Washington's influence within the Organization of American States to prevent Bolivian President Evo Morales from running for another term.
The presence of Russian, Chinese and Iranian planes in Venezuela has outraged the U.S at a time when it was looking to unseat the socialist-led Latin American country's leader.
The U.S. suspects that Russia sent specialists to help prepare the S-300 surface-to-air missile defense system for combat.
Russian troops will remain in the country "as long as they need it, and as long as the Venezuelan government needs them," Foreign Ministry spokesperson Maria Zakharova told reporters.
Two Russian air force planes landed near the capital city of Caracas with about 100 soldiers aboard, and U.S. officials believe the group included cybersecurity experts.
Senator Marco Rubio denied being behind the massive blackout in Venezuela, where President Nicolás Maduro vowed to defeat "American imperialism" trying to overthrow him.
The outage is reportedly affecting the capital Caracas and 15 of the nation's 23 states.
"There should be no Democrats on the fence on this issue," Fox News Business host Trish Regan argued.
China said that "the Venezuelan government has maintained calmness and restraint and has spared no effort to maintain domestic peace and stability and to effectively avoid large-scale bloody conflicts."
President Nicolás Maduro mocked the U.S. president's attempt to persuade Venezuela's military to support opposition leader Juan Guaidó and allow humanitarian aid into the country. "Who is the commander of the armed forces—Donald Trump from Miami?" he asked.
Cuba's foreign minister lashed out at the U.S., alleging it is preparing for "military aggression" against Venezuela.
"I don't understand why members of this committee or the American people should find any testimony that you give today to be truthful," Congresswoman Ilhan Omar told U.S. special envoy to Venezuela Elliott Abrams.
Last month, President Donald Trump's national security adviser was spotted with a notepad that said, "5,000 troops to Colombia."