Under Russia’s proposal, civilians would be allowed to leave if the city’s defenders laid down arms.
But Ukraine has refused, saying there was no question of it surrendering the strategic port city.
Around 300,000 people are believed to be trapped there with supplies running out and aid blocked from entering.
Residents have endured weeks of Russian bombardment with no power or running water.
Details of the Russian proposal were laid out on Sunday by Gen Mikhail Mizintsev, who said Ukraine had until 05:00 Moscow time (02:00 GMT) on Monday morning to accept its terms.
Under the plans, Russian troops would have opened safe corridors out of Mariupol from 10:00 Moscow time (07:00 GMT), initially for Ukrainian troops and “foreign mercenaries” to disarm and leave the city.
After two hours, Russian forces say they would then have allowed humanitarian convoys with food, medicine and other supplies to enter the city safely, once the de-mining of the roads was complete.
Russian Gen Mizintsev admitted that a terrible humanitarian catastrophe was unfolding there – and said the offer would have allowed civilians to flee safely to either the east or west.
In response to the offer, Ukraine’s Deputy Prime Minister Iryna Vereshchuk said Ukraine would not stop defending Mariupol.
“There can be no question of any surrender, laying down of arms,” she was quoted by Ukrainska Pravda as saying.
Earlier on Sunday, Pyotr Andryushenko, who is an adviser to the mayor of Mariupol, vowed the city’s defenders would fight on.
“We will fight until the last of our soldiers,” he said.
He told the BBC’s Newshour that Moscow’s humanitarian promises could not be trusted, and repeated unconfirmed claims made by Mariupol officials in recent days that Russian forces have been forcibly evacuating some of its residents to Russia.
“When they [Russian forces] say about humanitarian corridors, what do they really do? They really force evacuate our people to Russia,” Andryushenko said.
The BBC has not been able to verify these accusations.
Mariupol is a key strategic target for Russia and has seen some of the invasion’s fiercest fighting, with Russian forces so far failing to take the city from its defenders.
A catastrophe unfolds
By Hugo Bachega, BBC News, Lviv
“Hell on Earth”. That’s how Yaroslav Zhelezniak, a Ukrainian MP from Mariupol, described to me the situation in his native city. And it really is desperate.
Mariupol has been surrounded by Russian forces, which have prevented the creation of corridors to allow the distribution of humanitarian aid.
Around 300,000 residents are trapped inside, with no electricity, running water or heating. And as food and medical supplies run low, the crisis could get worse, with people going hungry and diseases spreading.
Residents spend most of their time in shelters and basements as Russia continues its unrelenting attack on the city, from land, air and sea, officials say.
Pictures show a city in ruins, with entire neighbourhoods devastated. The mayor, Vadym Boychenko, told me last week that over 80% of residential buildings had either been damaged or destroyed, a third of them beyond repair.
Bodies are being left in the streets as it’s too dangerous to get them. When they’re finally collected, some end up buried in mass graves, another symbol of the horror there.
Even a Russian general has acknowledged that a terrible humanitarian catastrophe is unfolding in the city. Unsurprisingly, he didn’t admit that his own forces were responsible for that.
Since the war began, authorities say at least 2,500 people have been killed in Mariupol although the true figure may be higher.
After last week’s destruction of a theatre where more than 1,000 people were sheltering, on Sunday authorities in Mariupol said that an arts school with 400 people inside has also been attacked.
Previous efforts to evacuate Mariupol’s civilians have been blocked by Russian fire, although local authorities say that thousands have been able to leave in private vehicles.
On Sunday, the Ukrainian deputy prime minister said 3,985 people had fled from Mariupol to Zaporizhzhia, adding that on Monday the government plans on sending about 50 buses to pick up further evacuees from the city.
President Volodomyr Zelensky has said the Russian siege amounts to a “war crime”.
“This is a totally deliberate tactic,” he said. “They [Russian forces] have a clear order to do absolutely everything to make the humanitarian catastrophe in Ukrainian cities an ‘argument’ for Ukrainians to cooperate with the occupiers”.
The location of the port city, on the Sea of Azov, makes it a strategic target for Russia, as it would help it create a land corridor between the eastern regions of Donetsk and Luhansk, controlled by Russian-backed separatists and Crimea, which was annexed by Russia in 2014.