Ukrainian soldiers are continuing to mount resistance against a Russian onslaught at the Azovstal steel plant amid reports a 'traitor' gave away secrets that led to an attack.
Russian troops stormed the city last month and destroyed large areas, but pockets of organised resistance remain under siege at the plant, refusing to surrender.
Last week, Ukrainian authorities said over 1,000 of its troops - many of them wounded and injured - were in the underground tunnels of the area, which has so far held firm against Russian occupation.
All civilians have been evacuated, and only Ukrainian soldiers - including wounded servicemen - remain.
Fighters of the Azov Battalion, a unit of the National Guard of Ukraine, are currently holed up in the battered Mariupol plant.
According to the Reuters news agency, Ukrainian soldiers had to fend off a Russian attempt to breach their system of tunnels after a 'traitor' gave away their top secrets.
The Ukrainian Centre for Strategic Communications said: "Yesterday, the enemy did not shy away from an attempt to storm the grounds. Our main goal is to block the exits of the system of underground tunnels, which the enemy was shown by a traitor.
"Despite this, our Defenders of Mariupol are trying to counterattack, risking everything, said Petro Andryushchenko, adviser to the mayor."
Watch: Russian army 'lose entire battalion' trying to cross river
Serhiy Volyna, commander of the 36th Separate Marine Brigade, created a Twitter account for the sole purpose of reaching out to Musk in the hope of getting some more help to the besieged plant.
"People say you come from another planet to teach people to believe in the impossible," he tweeted.
"Our planets are next to each other, as I live where it is nearly impossible to survive. Help us get out of Azovstal to a mediating country. If not you, then who?"
Volyna also posted an appeal to Facebook to contact the SpaceX founder, writing: "Help us get out of Mariupol".
Mariupol has been the site of a number of alleged atrocities since Putin ordered his troops to invade on 24 February.
Mariupol city council officials claimed a Russian airstrike hit the Mariupol Drama Theatre where hundreds of people had been sheltering on 16 March.
Russia's invasion of Ukraine has now entered its third month, but has so far been deadly for Putin's troops.
From February 24 to May 12, the total combat losses of Russian troops stand at around 26,650 personnel, the General Staff of the Armed Forces of Ukraine has said.