Ukraine is readying soldiers for a major Russian offensive that is expected in the coming months, likely before the summer begins, Ukrainian President Volodymyr Zelenskyy said Sunday.

Zelenskyy emphasized that foreign aid "has to end up in tangible weapon systems," specifically, crucial long-range artillery, to give Ukraine a chance at victory. His remarks came one day after the House passed a $60.8 billion aid bill for Ukraine, which will now go to the Senate and President Biden for final signature. 

In a separate interview Sunday morning on "Face the Nation," Sen. Mark Warner, a Virginia Democrat who heads the Senate Intelligence Committee, said that he hopes the U.S. will be able to send aid shipments "with that longer-range ATACMS" as soon as possible. Warner told "Face the Nation" moderator Margaret Brennan that shipments should go out quickly once the congressional aid package passes the Senate and reaches Mr. Biden, on Tuesday or Wednesday.

"I hope once the President signs ... that these materials will be in transit by the end of the week," said Warner. "It's clearly been the case that the Ukrainians morale has been great, but it's been undermined over the last couple of months, when they have been literally given out rationed bullets, eight to 10 bullets a day. And on artillery shells — Russians 10 to one — you can't underestimate that Ukrainians' grit, determination. But if they don't have the materials, they can't carry this fight to the Russians." 

Zelenskyy told "Meet the Press" moderator Kristen Welker that being adequately prepared means having the weapons and equipment needed to fight back against the impending "full-scale" attack, and receiving those supplies before it happens.

"We are preparing. The personnel is preparing. The soldiers are engaged in training," he said in translated remarks during the "Meet the Press" interview. "And I don't want to repeat myself, but we are waiting for the weapon systems because we want to have the brigade fully equipped. Some of them are exhausted. We need to replace them. But these new brigades, they have to have the equipment."

Ukrainian troops managed to stave off Russian advancements through the winter but struggled to contend with Russia's seemingly limitless stock of missiles and shells, which have decimated some places and allowed for growing occupation in the East. 

During an interview several weeks ago in an undisclosed, bombed-out building in far-eastern Ukraine, Zelenskyy told a CBS News team that his country's intelligence officials anticipated another major offensive from invading Russian troops in June, or, potentially even sooner, around the end of May. At the time, he pressed for support from Ukraine's allies, including the United States, to help the country defend itself. 

"And before that, we not only need to prepare, we not only need to stabilize the situation, because the partners are sometimes really happy that we have stabilized the situation," Zelenskyy said. "No, I say we need help now."

Zelenskyy told Welker on Sunday's "Meet the Press" that the Kremlin intends to mobilize 300,000 troops by June 1. But Russia is planning to seize the eastern city of Chasiv Yar much earlier than that, by May 9, he said. Near the occupied city of Bakhmut, Chasiv Yar is still controlled by Ukraine but has been ravaged by Russian firepower in recent months and could be the next city to fall, with pleas for reinforcement coming from the troops stationed there.

"I visited the region recently. I talked to the soldiers. The soldiers say that they lack equipment," Zelenskyy said Sunday. "They need to fight Russian reconnaissance drones, which essentially guide the artillery. And we need artillery shells. I hope we will be able to stay, and the weapons will come on time, and we will repel the enemy and then we'll break the plans of the Russian Federation with regards to this full-scale offensive."

Zelenskyy had noted on "Meet the Press" that U.S. aid would not only strengthen Ukraine in its defense against Russia, but also "send the Kremlin a powerful signal that it will not be the second Afghanistan" — which fell to the Taliban as U.S. troops withdrew in 2021 — "and the United States will stay with Ukraine, will protect the Ukrainians, and they will protect democracy in the world."

Sen. Dan Sullivan, an Alaska Republican who supports sending military aid to Ukraine, referred back to that point in another "Face the Nation" interview Sunday. 

"I believe strongly that the weakness coming out of the Biden White House ... and the botched withdrawal from Afghanistan emboldened Putin to undertake the invasion of Ukraine. So we've got to remember, kind of broadly, more strategically, where we are," Sullivan said. "But as you know, I was a supporter of the defense supplemental. And I think it's not just going to be important for Ukraine, as you and I have talked about before, it's going to be really important for the industrial base of America, which has atrophied dramatically."

Still, Sullivan did not respond definitively when asked whether former President Donald Trump would continue to stay with Ukraine if he were to be elected again in November. "Listen, I can't guarantee anything," he said.