"Ukraine proposes an end to this war through direct contact between us - and you. I propose a meeting," Zelenskyy writes in the letter, which was published on the presidential office's website.
"Ukraine is ready for a full ceasefire for the duration of negotiations," Zelenskyy added.
Zelenskyy has requested a meeting with Putin several times previously, but it is rare for him to address the Russian president directly.
A meeting must take place on neutral ground, according to Zelenskyy, who writes that "A Ukrainian leader has no business in your capital - just as a Russian leader has no business in Kyiv."
But in an initial comment from the Kremlin, spokesman Dmitry Peskov, quoted by state media, said briefly that Zelenskyy is welcome in Moscow "at any time."
Putin: Preferably a peace agreement
During a press conference earlier on Thursday with international journalists, Putin expressed confidence that Russia can achieve its goals in Ukraine militarily, but said he would rather see them fulfilled through a peace agreement.
However, this agreement - which Reuters says is based on an accord he reportedly reached with Donald Trump in Alaska last year - depends, according to Putin, on Ukraine's willingness to compromise.
"If you can't reach an agreement peacefully, you have to seek a military solution," Putin said.
"We want to ensure that there can be no armed forces in Ukraine that could pose a threat to Russia."
US President Donald Trump says a meeting between Zelenskyy and Putin would be excellent, but that both sides must compromise.
"I'm glad they might be talking about meeting. I think we had a lot to do with that," he said at a press conference in the Oval Office.
Improve air defense
However, Putin's words about achieving peace through agreements come just after he threatened to expand the use of the hypersonic Oreshnik missile against Ukraine.
The missile, which has been used three times in the war, can, according to Russia, be equipped with nuclear weapons.
After high-profile Ukrainian drone attacks against Russia, including against St. Petersburg on Wednesday in connection with the economic forum taking place there, Putin also says that Russia must strengthen its air defense.
"And we will do that."
