"We are ready to reach a deal, but there are still some specific points... which need to be fine-tuned, and we are busy with this," Sergei Lavrov said in an interview with CBS News.
Trump's envoy Steve Witkoff is due in Russia on Friday where he is expected to hold another round of ceasefire talks with Putin.
Lavrov said the talks process was moving in the right direction, and negotiations would continue with Washington.
He said the US president was "probably the only leader on Earth who recognised the need to address the root causes of this situation", but said Trump "did not spell out the elements of the deal".
Trump, however, issued a direct appeal to Russian President Putin following missile and drone strikes on the Ukrainian capital early Thursday which left at least 12 people dead.
It was the latest in a wave of Russian aerial attacks that have killed dozens of civilians, defying Trump's push to bring about a rapid end to the bloodshed.
"I am not happy with the Russian strikes," Trump said on social media. "Not necessary, and very bad timing. Vladimir, STOP!"
Trump, who is accused of favoring Russia and has often vilified Ukrainian President Volodymyr Zelensky, was asked by reporters what concessions Moscow had offered in negotiations to end the war.
"Stopping taking the whole country -- pretty big concession," he replied.
Russia launched its full-scale invasion of Ukraine in 2022, hoping to take the country in days, but has since been bogged down in a bloody war with huge casualties on both sides.
Crimea spat
Zelensky cut short a trip to South Africa to deal with the aftermath of the latest strikes.
He questioned whether Kyiv's allies were doing enough to force Putin to agree to a full and unconditional ceasefire.
"I don't see any strong pressure on Russia or any new sanctions packages against Russia's aggression," Zelensky said, highlighting that Trump had previously warned of repercussions if Moscow did not agree to pause the fighting.
Trump on Wednesday had accused Zelensky of frustrating peace efforts by ruling out recognizing Russia's claim over Crimea, a territory the US president said was "lost years ago". Moscow annexed the peninsula in 2014.
"We do everything that our partners have proposed; only what contradicts our legislation and the Constitution we cannot do," Zelensky said in response to a question about Crimea.
In contrast, NATO Secretary-General Mark Rutte said Thursday it was Moscow, not Kyiv, that needed to move forward in negotiations.
"The balls are clearly in the Russian court now," Rutte told reporters at the White House after meeting Trump.
'Pulled out of the rubble'
Russia fired at least 70 missiles and 145 drones at Ukraine between late Wednesday and early Thursday, the main target being Kyiv, the Ukrainian air force said.
"As of 5:30 pm (1430 GMT), the death toll in Kyiv's Sviatoshinsky district has risen to 12," Ukraine's state emergency services reported, with the number of wounded rising to 90.
Russia said it had targeted Ukraine's defense industry, including plants that produced "rocket fuel and gunpowder".
Asked about the strikes, Lavrov told CBS News: "We only target military goals or civilian sites used by the military."
"If this was a target used by the Ukrainian military, the Ministry of Defense, the commanders in the field have the right to attack them."
Ukraine has been battered by aerial attacks throughout Russia's three-year invasion but strikes on Kyiv, better protected by air defenses than other cities, are less common.
Zelensky said Russia used a North Korean ballistic missile in the strikes.
Olena Davydiuk, a 33-year-old lawyer in Kyiv, told AFP she saw windows breaking and doors "falling out of their hinges".
"People were being pulled out of the rubble," she added.
Zelensky said that on the ground, Russian forces had been attacking Ukrainian positions on Thursday, following the Kyiv strikes.
"Basically, the Russians tried to go on the offence under the cover of their massive strike," he said on X.
"While the bulk of our forces were focused on protection from missiles and drones, the Russians significantly intensified their ground attacks."
Source: AFP