Zelenskiy has tried to give momentum to a US-led peace effort, offering a ceasefire and a summit meeting with Russian President Vladimir Putin.
Moscow has taken issue with some of the proposals, however, and an end to the bloodshed appears no closer.
Additionally, international concerns have mounted recently that the fighting could spread beyond Ukraine's borders as European countries rebuked Russia for what they said were provocations.
The incidents have included Russian drones landing on Polish soil and Russian fighter aircraft entering Estonian airspace.
Latvian President Edgars Rinkevics said on social media that Russia was testing NATO's political and military response and aiming to reduce Western support for Ukraine by compelling countries to redirect resources toward the defence of alliance countries.
Calibrating how to respond to Russia was not easy, Rinkevics said Sunday.
Russia was doing just enough not to cross a red line, but things could still spiral, he said.
Russia's defence ministry denied any Estonian airspace violation by its planes, and Kremlin spokesman Dmitry Peskov on Monday dismissed Estonia and NATO's claims as "void, ungrounded and continuing an unbridled policy of inciting tensions and provoking confrontation".
Zelenskiy was due to attend the annual high-level gathering at the UN General Assembly, where he planned to recruit support for efforts to stop Russia's invasion.
"The schedule already includes nearly two dozen meetings with leaders from different countries, from all parts of the world," Zelenskiy said on Telegram.
Zelenskiy said he also planned to meet with US President Donald Trump, whose drive for a peace deal after taking office in January has yielded no progress.
"It is vital that this week strengthen the world's resolve for robust action - for without strength, peace will not prevail," Zelenskiy said.
He said that in the past week Russia fired more than 1500 strike drones, 1280 glide bombs and 50 missiles of various types at Ukraine.
More than 132,000 foreign components were found in those weapons from dozens of countries, he said.
Ukraine has campaigned for tighter sanctions on Russia.
Meanwhile, at least seven Russian aircraft bombarded the southern Ukrainian city of Zaporizhzhia overnight, killing three people and wounding two others, according to regional administration head Ivan Fedorov.
The attack started about 4.20am on Monday and lasted about 40 minutes, Fedorov said.
Residential buildings, shopping centres, a car park and "critical infrastructure" were targeted, he said, none of which "had anything to do with military infrastructure".
The Ukrainian air force said it stopped 132 out of 141 strike and decoy drones launched by Russian forces overnight.
Russia made similar claims. The Moscow-appointed head of Ukraine's Russia-occupied Crimea peninsula, Sergei Aksyonov, said three people were killed and 16 others were wounded late on Sunday by Ukrainian drones that struck the popular vacation resort of Foros.
The Russian defence ministry said there are no military facilities there.
In Russia's Belgorod border region, three people were killed and another 10 were wounded by Ukrainian drone attacks on Sunday, according to the regional governor, Vyacheslav Gladkov.
The defence ministry said 114 Ukrainian drones were downed early on Monday over several Russian regions.
