The yesterday's advance came as Moscow and Washington traded barbs over stalled efforts to end fighting in the city, where forces loyal to President Bashar al-Assad have made significant advances since last week.
But US Secretary of State John Kerry said he would try to get Syrian peace talks going again with Russia's help.
On the humanitarian front, German Chancellor Angela Merkel said it was a "disgrace" that the international community had been unable to alleviate the suffering in east Aleppo.
The monitor called Shaar "the most important neighbourhood in the heart of east Aleppo", and said rebels were being reduced to fighting a "war of attrition".
The rapid regime gains have left opposition fighters scrambling to defend the shrinking enclave they still control in Aleppo's southeastern districts.
Despite mounting criticism of the offensive begun on November 15, world powers have struggled to find a way to halt the fighting.
"Russia says Assad is ready to come to the table... And I am in favour of putting that to the test," he said.
Kerry, who has had repeated meetings on Syria with his Russian counterpart Sergei Lavrov, said they would meet again today or tomorrow in Hamburg, Germany.
Even if Aleppo falls, "the war will continue," Kerry said, stressing that there had to be a negotiated settlement.
But yesterday, Lavrov accused Washington, which has backed rebel groups against Assad, of backtracking.
"It looks like an attempt to buy time for the rebels to have a breather, take a pause and replenish their reserves," Lavrov said. Moscow had the impression that "a serious discussion with our American partners isn't working out".
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