That poor run, along with the ongoing investigations into alleged racist remarks by skipper John Terry, have created a gloomy mood at Stamford Bridge where the home fans were stunned by Liverpool's late goal from former Chelsea defender Glen Johnson which condemned them to a 2-1 defeat.
However Villas-Boas, 34, who took over at Chelsea after winning the treble with Porto last season, said he did not fear losing his job despite becoming the first Chelsea manager since Claudio Ranieri in 2002 to lead his side to successive home league defeats following last month's 5-3 loss to Arsenal.
Earlier the Sunday Express reported that Abramovich, who has fired six managers since buying the club eight years ago, was unimpressed with Chelsea's start to the season which has left them 12 points behind leaders Manchester City after 12 games.
"He didn't pay 15 million to get me out of Porto and doesn't want to pay another fortune to get me out of here," Villas-Boas told reporters.
"It is not a question of the owner losing patience. We have set out to build something new at this club and the club is committed to what we are building for the future."
However, he did admit his side now face an uphill task to win the title following recent defeats by Queens Park Rangers, Arsenal and Liverpool.
"It is not good to be so low in relation to the leaders at this stage of the season," he said.
"But our commitment is to the club and to what we are doing in the future and we have enough talent to compete in all competitions. This our perspective at the moment."
Since Abramovich bought Chelsea in 2003, Ranieri, who was already manager when the Russian arrived but got fired in 2004, Jose Mourinho, Avram Grant, Luiz Felipe Scolari, caretaker Guus Hiddink and Carlo Ancelotti have all come and gone.
Chelsea Chairman Bruce Buck said last month that Villas- Boas would be manager for the next 15 years, but some wags have suggested he will be lucky to last another 15 minutes unless results improve soon.
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Even Ancelotti, who led Chelsea to the FA Cup and Premier League double in 2010, was gone 12 months later because they did not win a major title last season.
"There is no running away from our responsibilities," said Villas-Boas. "There is no calling this a year of transition, no calling for time to do our work.
"Our responsibility is to win the most amount of trophies we can and at the moment we are in four competitions.
"It is not the brightest of starts in the Premier League for Chelsea in the last 10 years, but the belief is there from the team and that is what we have to focus on."
With former Chelsea caretaker Hiddink back in the job market after leaving his position as Turkey coach, London bookmakers William Hill rate Villas-Boas as a 9-1 shot to be the first Premier League manager to be fired this season.
"It is not impossible for us to turn it around, and the Decem
