But Trump's requests do not seem feasible, and the EU notes that it has already hammered the Kremlin with sanctions.
Meanwhile, diplomats fear it could be a ploy by Trump to again stall on taking a tough stance against Russia himself.
- Stopping oil purchases? -
In a Truth Social post over the weekend, Trump said he would impose "major sanctions" on Moscow if all NATO countries stopped buying Russian oil.
The 27-nation EU has already banned most imports of Russian oil after the Kremlin's 2022 invasion of Ukraine, slashing the share of oil it imports from Russia from 29 percent in early 2021 to two percent by mid-2025.
Currently Hungary and Slovakia, both countries friendly to both Moscow and Trump, still buy oil from Russia.
The EU is planning to end that entirely and has announced a plan to phase out purchases of all Russian oil and gas by the end of 2027.
EU chief Ursula von der Leyen has said Brussels is "looking at phasing out Russian fossil fuels faster" as it works with partners on toughening sanctions against Moscow.
Other officials have suggested they plan to stick broadly to a timeline EU energy commissioner Dan Jorgensen called "very ambitious" after talks with his US counterpart last week.
But diplomats have welcomed any possible pressure from Trump on Budapest and Bratislava, arguing it could help stiffen European resolve.
"We wouldn't mind some extra push on Hungary, Slovakia from his side as well," an EU diplomat said, speaking on condition of anonymity to discuss sensitive deliberations.
If Trump is serious about getting all NATO countries to turn off the taps, then the bigger issue could be non-EU member Turkey.
Ankara has refused to join international sanctions on Moscow and has even stepped up its purchases of Russian oil.
- Tariffs on China? -
If Trump is largely preaching to the choir on trying to get Europe to cut Russian oil imports, then on China tariffs he is swimming against the tide.
In his post he said NATO countries should place "50% to 100% TARIFFS ON CHINA, to be fully withdrawn after the WAR with Russia and Ukraine is ended".
The EU has already blacklisted firms and banks in China accused of supporting Russia's military or helping circumvent EU sanctions.
But the free-trading bloc is no fan of tariffs, and while it has its own grievances over China's commercial practices, Brussels has no appetite for a broader trade war with the Asian giant.
"Tariffs is not something that is really discussed at the moment," the EU diplomat said.
- So what is the EU doing? -
Brussels is about to put forward its proposals for a 19th package of EU sanctions on Russia since the invasion of Ukraine.
Despite the bloc's top sanctions envoy holding talks in Washington last week, diplomats say that genuine coordination with the United States does not seem on the table.
Officials say the next round of measures should see more Chinese businesses targeted, but they do not expect a broader onslaught against Beijing.
Sceptical voices have also pointed out that Trump's demands for action coincide with US commercial interests.
But a second diplomat said the requests put the bloc "in a tight spot".
"Even if his requests are deliberately excessive, it still forces us to come to terms with them in some kind of way in order to avoid him shifting the blame onto the EU," the diplomat said.
By demanding something he knows the EU is not willing to do, officials worry that Trump could be looking to justify not going after Russia himself.
"What is dangerous is that the US administration hints that if we don't follow through with this idea, the EU is not serious enough in the US's eyes in wanting to end the war," a third EU diplomat said.
"This would be a dangerous spin."
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