The Daily Mail says Abramovich is already understood to be considering the possibility of looking for his seventh manager since taking over in 2003 if the Italian fails to respond to a huge setback by not landing a third English championship.
Ancelotti admitted on Tuesday night that losing to his most prominent predecessor had put him in a precarious position.
"I've worked for a long time in this environment and I know exactly what the expectations are," he said.
"It's normal that a manager is going to come under pressure if he loses a few matches."
Abramovich turned to Ancelotti because he had won the Champions League twice as AC Milan manager. He did not expect the Italian to fail to get beyond the last 16.