U.S. Secretary of State Marco Rubio said Sunday the United States is not at war with Venezuela, adding that the U.S. is "running policy" in Venezuela — not with troops on the ground but through financial leverage on the remaining allies of former-President Nicolas Maduro. 

"We want Venezuela to move in a certain direction because not only do we think it's good for the people of Venezuela, it's in our national interest," Rubio said in an interview on NBC's Meet the Press. 

Rubio, who is also President Trump's interim national security adviser, said the U.S. will enforce its influence through an existing "oil quarantine" imposed by U.S. naval vessels near the coast of Venezuela.

"Our military is helping the Coast Guard conduct a law enforcement function, which is not just the capture of Maduro, but the enforcement of our sanctions," Rubio said on the program. "And we think that's just tremendous leverage, incredible crippling leverage, which we intend to continue to use until we see the changes that we need to see." 

Since September, U.S. forces have struck at least 35 alleged drug boats, killed at least 115 people on board and seized oil tankers near Venezuela. 

The comments offer the first glimpse of what U.S. policy toward Venezuela will look like following the weekend raid that captured the country's leader, and Trump's remarks that the U.S. will "run" Venezuela.

Senate Intelligence Committee Chair Tom Cotton, R-Ark., echoed Rubio in calling for a pressure campaign against Maduro's allies who still control the country. Cotton specifically named interim President Delcy Rodríguez, who previously served as vice president under Maduro.

"They have control of the military and security services," Cotton told CNN. "We have to deal with that fact."

During a Sunday interview with The Atlantic magazine, Trump said Rodriguez would "pay a very big price" if she didn't cooperate, a day after he claimed that she was on board with the U.S. plans.

The Trump's administrations actions over the weekend will likely see the first signs of domestic pushback this week in the form of congressional Democrats.

Senate Minority Leader Chuck Schumer, D-N.Y., condemned Trump for ordering the Saturday military operation without consulting Congress. The administration argues that it didn't need to notify Congress because Maduro was arrested in a law enforcement action. The Venezuelan president will appear before a New York judge Monday.

"They went inside Venezuela, bombed civilian as well as military places," Schumer said. "And it's a violation of the law to do what they did without getting the authorization of Congress."

Schumer said Sunday he plans to hold a vote on a war powers resolution this week that would limit Trump's ability to attack Venezuela without Congressional approval. 

"If it's voted positively in both houses, then the president can't do another thing in Venezuela without the OK of Congress," Schumer said.

Senator Rand Paul, R-Ky., is the only GOP member publicly supporting the resolution so far. Unlike other measures, this resolution needs a simple majority to pass the Senate — meaning only three more Republicans would have to support it to advance the measure. It is unclear if the measure has Republican support beyond Paul.