"Bolton is here to underscore US support for Ukraine's sovereignty, territorial integrity and Euro-Atlantic path," the US Embassy said on social media.
US President Donald Trump's advisor was expected to hold talks with Zelensky on Wednesday.
"We support President Zelensky's reform efforts and vision to create a stronger and more prosperous Ukraine," Bolton wrote on Twitter.
After arriving, Bolton laid flowers at a memorial for those killed in the war against Russia-backed separatists in the east of the country.
Trump could meet Zelensky in Poland in the coming days, Bolton told reporters.
"We hope that the presidents will have the opportunity to talk in Warsaw as part of events marking the 80th anniversary of the outbreak of World War II," the website Evropeiska Pravda quoted Bolton as saying in comments translated into Ukrainian.
Both Zelensky and Trump are set to travel to Poland for the commemorations on Sunday.
Speaking later to the Ukrainian service of the US radio channel Radio Free Europe/Radio Liberty (RFE/RL), Bolton said: "President Zelensky would be well advised to look at how to unfold a strategy of dealing with the Russians very carefully."
"I don't think there's any reason to rush into one course of action or another," he added.
Asked about relations between Moscow and Washington, Bolton said: "The Trump administration has been very tough. It has imposed enormous sanctions on Russia."
Bolton said he would warn the Ukraine leader about the dangers of cooperation with China after the Wall Street Journal reported last week that the US aims to block China from acquiring Ukrainian aerospace company Motor Sich.
"Military and sensitive technologies should not reach opponents or potential opponents," Evropeiska Pravda quoted Bolton as saying, while he declined to comment specifically on Motor Sich.
"We inform friends and partners about the dangers that Chinese investments bring," he added.
Bolton's visit came after French President Emmanuel Macron announced Monday that the leaders of Russia, Ukraine, Germany and France would hold peace talks on the conflict in eastern Ukraine in September, but did not give an exact date.
Ukraine has been locked in conflict with Russia-backed separatists since Moscow annexed Crimea in 2014, a move the international community refuses to recognise.
Some 13,000 people have been killed in the violence that Kiev blames on Russian President Vladimir Putin.
Moscow has denied claims it has funnelled troops and arms across the border.
The last Ukraine peace summit between the four countries was held in 2015.