The Mail on Sunday says having lost the World Club Cup and been knocked out of the Capital One Cup by Swansea, Benitez would need to win the FA Cup or Europa League to convince owner Roman Abramovich to give him the job.
But Benitez believes his side are making progress and the key figures at the club - Abramovich, his assistant, Marina Granovskaia, and technical director Michael Emenalo - are said to be impressed with his approach to training and tactics.
"I'd like to be in a top side," he said. "And Chelsea are a top side with great potential. So it's a great opportunity. I came in the middle of the season, so I couldn't change too many things and I had to work with the squad I had.
"This team have enough quality and they're a good squad but the circumstances of this year have been difficult. This is a top side and if they continue doing things in the right way they will always be challenging for trophies."