Mourinho was subject of the first managerial vote of confidence of Abramovich's 12-year ownership on October 5, and Chelsea have won just one of four matches since ahead of Saturday's clash with Liverpool.
Chelsea, the defending Barclays Premier League champions, enter this weekend's fixtures 15th in the table after losing half of their 10 games, including last Saturday at West Ham, while their Capital One Cup defence ended at Stoke on Tuesday.
Asked whether he had been given assurances over his position, Mourinho, who in August signed a four-year contract extension until June 2019, said: "I don't have to tell you."
Abramovich has previously axed managers when Chelsea's place in the Premier League's top four and qualification for the Champions League the following season had been in doubt.
Asked if he can guarantee Chelsea a place in Europe's elite competition next term, Mourinho said: "I cannot promise."
He then pointed to the failure of Manchester United and Liverpool to qualify for the Champions League in recent seasons.
Mourinho added: "Chelsea is a big club
It's the club that I choose to come (to)
It's a club I live in a very specific way every day.
"Liverpool is a big club, Man United is a super big club
Last season Liverpool win nothing and didn't qualify for the Champions League.
"Two years ago Man United didn't qualify for the Champions League - not even Europa League - and won nothing.
"They are still a monster club, but that's football."
Mourinho declined to comment on two disciplinary matters hanging over him.
The 52-year-old Portuguese had until Thursday at 6pm to respond to a misconduct charge relating to his half-time dismissal by referee Jonathan Moss at West Ham.
Mourinho expects his appeal against a Â£50,000 fine and suspended one-match stadium ban for comments made about referee Robert Madley following the October 3 loss to Southampton to be heard on Friday.
"I think so," he said
"Not optimistic, not pessimistic - just wait."
Source : PA