Kyiv is under pressure from the White House to secure a quick peace but is pushing back on a US-backed plan proposed last month that is widely seen as favourable to Moscow.
Ukrainian officials are also seeking strong security guarantees from partners, in the event of a deal, to prevent Russia from attacking again in the future.
Zelenskiy said new components of the deal hashed out with the British, French and German leaders in London on Monday were "more developed" and ready for US review.
"The Ukrainian and European components are now more developed, and we are ready to present them to our partners in the US," he wrote on X.
"Together with the American side, we expect to swiftly make the potential steps as doable as possible.
Finnish President Alexander Stubb said on Tuesday that allies have been working on three separate documents, including a 20-point framework, a set of security guarantees and a reconstruction plan.
"I think we are closer to a peace agreement than we have been at any time since the war began," he said at an event in Helsinki.
Stubb spoke as US President Donald Trump piled new pressure on Zelenskiy to secure a deal that could involve painful concessions, citing Russia's "upper hand" as it advances on the battlefield.
Among other demands, Russian President Vladimir Putin has said Ukraine must hand over its entire eastern Donbas region before Russia stops fighting, something which Zelenskiy has consistently rejected.
"They're much bigger. They're much stronger in that sense," Trump said in an interview with Politico.
He added that Zelenskiy would "have to get on the ball and start... accepting things".
Zelenskiy said he was prepared to hold elections within three months if the US and Kyiv's other allies could ensure the security of the vote.
Wartime elections are forbidden by law but Zelenskiy, whose term expired last year, is facing renewed pressure from Trump to hold a vote.
"I'm ready for elections, and moreover I ask ... that the US help me, maybe together with European colleagues, to ensure the security of an election," Zelenskiy said to reporters.
"And then in the next 60-90 days Ukraine will be ready to hold an election."
Zelenskiy's remarks followed comments by Trump suggesting Ukraine's government was using the war as an excuse to avoid elections.
"You know, they talk about a democracy, but it gets to a point where it's not a democracy anymore," Trump said.
Zelenskiy dismissed suggestions that he was clinging to power as "totally inadequate".
