The Syrian regime extended its grip on Aleppo to almost all of the city on Monday as U.S. and Russian negotiators were at an impasse over an American proposal to evacuate rebels and civilians from the few remaining opposition-held districts.

Syrian state media said President Bashar al-Assad’s forces had captured 98% of the eastern areas of Aleppo that have been largely rebel-held since 2012. The rebels and the opposition monitoring group Syrian Observatory for Human Rights said regime forces were in control of 90%.

Rebels...