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.
