Zelensky confirmed the pair talked about the issue during a meeting at the prime minister's country retreat northwest of London.
Afterwards he said he was "very positive" about creating a "jets coalition" for the war against Russia and that a decision would come "in the closest time".
"You will hear some I think very important decisions," he said in English but added that "a little bit more" work was needed until then.
Zelensky in February gave a much-acclaimed speech to the UK parliament in which he pleaded for Western jets, on only his second trip overseas since the war began.
His tour of European capitals starting last week has helped Ukraine secure more promises from France and Germany of military aid before an expected Ukrainian counter-offensive.
But NATO members have so far drawn a line at sending Kyiv modern fighter jets.
Nevertheless, Sunak confirmed that Britain would help Ukraine to train "combat-ready pilots" as part of its support package.
"We're ready to implement those plans in relatively short order," he told reporters.
Sunak said he would talk to other countries' leaders about aircraft support at the upcoming Council of Europe and G7 meetings this week.
His Downing Street office meanwhile confirmed that Britain would in the coming months send "hundreds" of both air defence missiles and long-range attack drones with a range of more than 200 kilometres (125 miles) to fend off Russia's invasion.
Last week, Britain became the first Western country to offer long-range cruise missiles to Ukraine, with its Storm Shadow rockets.
Dressed in his trademark military fatigues, Zelensky gave a bear hug to Sunak when he stepped off a Royal Air Force Chinook helicopter in the grounds of Chequers.
Before the talks, Zelensky tweeted that Britain was a "leader when it comes to expanding our capabilities on the ground and in the air".
Sunak pledged Britain's long-term security support for Ukraine.
"The frontlines of (Russian President Vladimir) Putin's war of aggression may be in Ukraine but the fault lines stretch all over the world," he said.
"It is in all our interest to ensure Ukraine succeeds and Putin's barbarism is not rewarded."
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