Ukrainian leaders on Monday pressed Capitol Hill lawmakers to back a no-fly zone over the country, saying Russian planes are already attacking schools, hospitals and other civilian targets. 

The leaders told more than a dozen members of the House Ukrainian Caucus that casualties will only mushroom if the international community doesn't step in to seize Ukrainian air space from Moscow's military forces.

"If we do not have a no-fly zone over Ukraine ... they will probably erase Kyiv, Kharkiv and the other cities from the map," Oleksandra Ustinova, a member of Ukrainians parliament, told reporters after the closed-door meeting.

"The first day they shot the air bases, they shot the airports, and the military bases. Now they're shooting kindergartens, children's hospitals," she continued. "They want to basically destroy the country ... if the international community does not protect the sky."

The White House earlier on Monday had said a no-fly zone was a bad idea that would bring the U.S. closer to a military conflict with Russia itself. 

A number of lawmakers attending the meeting also immediately rejected the proposal, warning it would require U.S. fighter pilots to engage directly with the Russian forces in a confrontation that many fear could escalate into a much deeper conflict between two of the world's premier nuclear powers.

Russian President Vladimir Putin has already shifted his nuclear arsenal to a state of "special combat readiness," an escalatory step the Biden administration has suggested is posturing and meant to ratchet-up tensions.

"Obviously the Ukrainians would like us to do everything," said Rep. Brad Sherman (D-Calif.). "That includes a no-fly zone, but frankly a no-fly zone means dog fights between American and Russian pilots, and that is dangerous."

Sherman, a senior member of the House Foreign Affairs Committee, is pushing several other strategies for assisting the Ukrainians in the air. One proposal, promoted by the Ukrainian leaders in Monday's meeting, would have the United States fly surveillance planes — known as AWACs, referring to the onboard Airborne Warning and Control Systems — over Poland to provide information about Russian troop movements to the Ukrainian military.

Sherman also proposed to have aircraft, flying under the banner of the United Nations, deliver food and medical supplies directly to Ukrainian cities.

"There's going to be food shortages. There's going to be medical shortages. And I want to see whether Putin threatens to shoot down U.N. planes flying only medicine and food into Ukraine," he said. "This is a man who claims that Ukrainians are Russians, that these are his people. Does he want to starve them? Does he want them to die from lack of medical care?"

Ukrainian ambassador to the U.S. Oksana Markarova was also on hand, praising the U.S. for helping to beef up Ukraine's military forces and applying economic sanctions on Russia's financial sector and wealthy elites. But she also pressed lawmakers to go further in all areas.

"We can continue to do more on all of these fronts," she said. "We need more weapons; I mean, we're not asking anyone to fight for us. We are defending our country ourselves. But we need all the support that all the civilized world can give us."

Congress is poised to step in with billions of dollars more in military and humanitarian spending. The Biden administration has said it will request $6.4 billion in additional aid, and leaders in both parties are lining up to support it.

"When you're dealing with this kind of threat, hope and prayers matter," said Rep. Mike Quigley (D-Ill.), a co-chair of the Ukraine Caucus. "But when you're dealing with Putin, hope and a Stinger and a Javelin are even better."

Ustinova, the member of parliament, said she's holding out hope that Ukraine's western allies will eventually back a no-fly zone. She said there was some support for the concept among congressional lawmakers during Monday's meeting. And she noted that there was staunch opposition to the idea of barring certain Russian banks from the international SWIFT system — until that opposition eroded over the weekend and those sanctions were imposed.

"We can fight on the ground. Ukrainians have proved we will fight to the very last bullet. We will fight bare-handed," she said. "But we need to protect the sky because they are using the sky to shoot us."