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President Trump pushed Russia and Ukraine to agree to halt strikes on each other’s energy infrastructure, as the White House continued to face bipartisan criticism over moves in recent days to ease U.S. pressure on Moscow and shift blame for the continued fighting to Kyiv.

“Effective immediately, there is an ENERGY CEASEFIRE in the War between Russia and Ukraine. Both parties have agreed,” Trump posted on Truth Social on Sunday, despite Moscow’s rejection of such a deal so far and no sign that there was an agreement in place.

The latest Trump statement came as he encountered widespread pushback on Capitol Hill to his decision to allow Russia to export oil to the U.S. and world markets and his harsh remarks toward Ukraine President Volodymyr Zelensky . On Sunday, a group of Democratic and Republican senators issued a joint statement calling Trump’s export plan illegal and contrary to a 2022 law that banned Russian energy imports.

Trump is moving quickly to try to ease energy prices with the U.S. midterm elections just weeks away, with high gasoline and diesel prices driven by the Iran war and the Russia-Ukraine conflict contributing to voter discontent toward Republicans. But analysts said the impact of new Russian supplies on lowering prices would likely be minimal.

Russian President Vladimir Putin has so far rebuffed proposals for an energy ceasefire and the Kremlin didn’t immediately respond to a request for comment. Trump previously announced such a ceasefire on Sept. 14, but Russia and Ukraine continued their strikes.

Zelensky reiterated on Sunday that he is open to stopping airstrikes against Russian oil refineries if Moscow stops bombing Ukraine ’s energy grid, but has staved off Trump’s pressure for Kyiv to cease its strikes unilaterally. He said Ukraine’s skies have been “infested” with Russian drone and bomb strikes throughout the day.

“Ukraine wants peace. The question is whether Russia does,” Zelensky said in a social-media post. “We are awaiting clear details of the dialogue between America and Russia – and we expect Putin to follow through on what President Trump said.”

Democrats and some Republicans have objected to Trump’s efforts to ease economic pressure on Russia, which launched its full-scale invasion of Ukraine in February 2022.

The U.S. “should not provide Putin with a financial lifeline while Russia continues to kill Ukrainian civilians. The administration must immediately reverse course,” a bipartisan group of senators said in a statement on Sunday. The group comprises Republican Sens. Susan Collins of Maine, Lisa Murkowski of Alaska, John Curtis of Utah and Thom Tillis of North Carolina and Democratic Sens. Jeanne Shaheen of New Hampshire, Ron Wyden of Oregon and Richard Blumenthal of Connecticut.

The senators said the administration failed to consult with Congress or submit the justification required under the 2022 law barring Russian energy imports to the U.S.

The White House didn’t respond to a request for comment on the senators’ statement.

Russia continued to bomb Ukraine’s energy grid and other targets in Kyiv on Sunday, with explosions audible throughout the day and power cuts continuing.

French President Emmanuel Macron , expressing support for an energy truce in a post on X, said: “At this stage, Russia continues to refuse it by intensifying its strikes against Ukrainian infrastructure in recent hours.”

Trump’s statement Sunday came after he lambasted Zelensky on Saturday, calling for his removal as Ukrainian leader and blaming him for Russia’s continuing war on Ukraine—as well as for pushing up fuel prices, which have risen globally since the U.S.-Iran war disrupted Middle East energy exports.

With no end to the conflict with Iran in sight, Trump has presented Russia as part of the solution, announcing a deal with Putin on Friday to boost the supply of Russian diesel to the U.S. and world markets. Energy experts say the deal, if implemented, could benefit the Kremlin’s coffers but is unlikely to provide significant relief to global energy markets.

“Our president is trying to find every answer possible,” said Rep. Tom Emmer (R., Minn.) on ABC News on Sunday, defending the deal on Russian diesel.

Democrats saw it as a last-ditch effort to bring down prices ahead of the election.

“Literally, Trump has made a deal with the devil. He will do anything for Republicans to win” the midterms, said Rep. Donald Norcross (D., N.J.) on X.

Russia has been planning to resume exports of diesel anyway, said Janis Kluge, head of Eastern Europe research at the German Institute for International and Security Affairs, a think tank in Berlin. “There was no concession from Putin in this deal,” he said.

Extra Russian diesel is unlikely to relieve the stress in global fuel markets. “The Middle East is much more important,” he said.

Trump’s comments in recent days have shown the limits of U.S. leverage in the war between Russia and Ukraine, as Europe, rather than the U.S., has become the mainstay of support for Kyiv’s defense against Moscow’s invasion.

Ukraine has greatly expanded its domestic production of drones and other weaponry, while relying on European allies for financing and military aid. Trump has ended nearly all U.S. aid for Ukraine’s defense, but the U.S. continues to share valuable satellite and other intelligence with Kyiv.

“The Ukrainians still see the U.S. as a critical ally, but they’ve gotten themselves to a point, especially through their partnerships with Europe, where they cannot be crippled by the Trump administration’s policy swings,” said Shelby Magid, a deputy director at the Atlantic Council, a Washington think tank.

Bombing Russian refineries is the strongest card that Ukraine holds to force Putin to stop the war he launched nearly five years ago. Kyiv intends to keep playing it, Ukrainian officials have said in recent days—unless Putin changes tack and accepts a mutual moratorium on energy strikes.

Russia is bombarding Ukraine’s energy infrastructure with missiles and drones in an attempt to cut off heating, electricity, water and other essential services in Ukrainian cities ahead of winter.

In the past week, Russia has attacked Ukraine’s electricity system 196 times and its oil and gas infrastructure 20 times, Zelensky said in an interview with ABC News broadcast on Sunday. Russia’s missile and drone strikes on cities have also killed 153 people over that period, he said.

“It’s a signal that Russia doesn’t feel real pressure. They want to continue,” Zelensky said. “If they want [an] energy ceasefire, we are open for this.”

Write to Marcus Walker at Marcus.Walker@wsj.com and Robbie Gramer at robbie.gramer@wsj.com