Having defended his players following defeats this season, Conte publicly accused them of lacking motivation and hunger against Roma - two qualities that were the trademarks of last season's Premier League title success.
The relationship between Abramovich and Conte is said to be at an all-time low, with suggestions the Italian may be removed from his post.
Tuesday night's shambolic 3-0 Champions League defeat against Roma didn't help matters, either, with the Blues turning in arguably their worst performance of the Conte era as they slumped to an emphatic loss in the Italian capital.
The west London side showed some defensive solidarity with their first clean sheet since a 4-0 win at Stoke in September when Eden Hazard scored the victor in a 1-0 victory at Bournemouth last weekend.
The Chelsea boss believes the lean patch will test the character of his squad as it is easy to play good football when things are going well but it becomes a lot harder when questions are being asked. "But I think sometimes this can happen", he added.
"This season we are struggling".
"I would like to see him at Chelsea for as long as possible". But in this season we are struggling a lot because we are facing a lot of problems. My mission is to try to find the best solution.
The Blues, who are now fourth in the Premier League table, must bounce back when they host United this weekend.
"But this year because of the Champions League we are not having that much time because every three days, you can not play on the Wednesday, play Saturday and what?"
UCLA's Josh Rosen Ruled Out for Saturday's Clash
Rosen (concussion) did not travel to Utah for Friday's game against the Utes , Ben Bolch of the Los Angeles Times reports. Starting quarterback Josh Rosen did not make the trip to Salt Lake City , as he has been ruled out due to a concussion.
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