At a meeting in Brussels, the envoys assessed whether Hungary might lift its veto on the 90-billion-euro ($A148 billion) loan package, which Ukraine desperately needs to prop up its war-ravaged economy and help keep Russian forces at bay.
Hungary has insisted it must start receiving supplies of Russian oil again via Ukraine before it will unblock the funds.
Hungary and Slovakia rely on Russian oil to meet their energy needs.
They have both accused Ukraine of failing to repair a damaged pipeline that ships the oil.
Ukraine and most of its European backers oppose imports of Russian oil, which have helped to fund President Vladimir Putin's war, now in its fifth year.
In a post on social media, President Volodymyr Zelenskiy said on Tuesday that Ukraine had completed repairs on the Druzhba pipeline.
He said it "was damaged by a Russian strike" but "the pipeline can resume operation".
But outgoing Hungarian Prime Minister Viktor Orban has signalled he would approve the Ukraine loans only once the oil starts flowing again, so the envoys are awaiting a clear signal from Budapest that his veto will be lifted.
Orban, who has repeatedly blocked EU aid to Ukraine, lost an election on April 12 and is due to leave office in May, to be replaced by the pro-European opposition leader Peter Magyar.
Cyprus, which holds the EU's rotating presidency, intends to launch a written procedure to approve the final piece of the puzzle in the loan package.
That would require Hungary or any other objecting nation to state in writing why they oppose it.
Such procedures are often left open for 24 hours, sometimes more, and it was not immediately clear what time frame Cyprus would use.
It ultimately means that final approval could come on Thursday, when EU leaders meet for a summit in Cyprus.
Given the many false dawns in recent months, EU foreign policy chief Kaja Kallas was reluctant to speculate on the outcome when quizzed by reporters on Tuesday.
"We expect an agreement in 24 hours, so I don't want to jinx it," she said.
The 27-nation EU had originally intended to use Russian assets frozen in Europe as collateral for the loan, but that option was blocked by Belgium, where the bulk of the frozen assets are held.
In December, the Czech Republic, Hungary and Slovakia agreed not to stop their EU partners borrowing the money on international markets as long as the three countries did not have to take part in the scheme.
But Orban angered the other 24 countries by later reneging on that deal over the pipeline dispute and as campaigning heated up before the election that he lost in a landslide.
In a Tuesday address, Zelenskiy said "there can be no grounds for blocking" the loans any more.
"The EU asked Ukraine to repair the Druzhba oil pipeline, which had been destroyed by Russia. We have repaired it."
The EU has also been trying since February to push through new sanctions against Russia, which Hungary and Slovakia have been blocking.
The measures could take longer than the loan to approve.
Slovak Foreign Minister Juraj Blanar said on Tuesday his country would only agree once "Russian oil arrives in Slovakia through the Druzhba pipeline".
Economy Minister Denisa Sakova said Slovakia expected oil supplies to resume early on Thursday.
