Pellegrini revealed on Saturday that he and several players would be leaving the Spanish club at the end of this season due to its money problems, which have resulted in a ban from future European competition under UEFA's financial fair play rules.
"Everyone would prefer to stay but unfortunately, the circumstances we are in don't allow that," the Chilean said.
Mancini helped City end a long wait for a trophy when he won the FA Cup in his first season in charge, but Saturday's defeat could be the end of the road for the 48-year-old after a difficult season in which he sold problematic striker Mario Balotelli -- who he had signed in 2010 from his former club Inter Milan -- and failed to make it past the group stage in the Champions League.
"This speculation is rubbish. If I said it's rubbish, it's rubbish ... speculation and rubbish," he angrily told UK match broadcaster ITV.
Mancini said he was "very sorry" for the 40,000 City fans who traveled to London for the match, which was played in pouring rain.
"They scored in the last minute so for us it was difficult. But we didn't play very well -- I don't know why," he told ESPN.
"It was a difficult season because our target was to win the Premier League, and we didn't win it.
But this can happen, it's impossible to win always.
"And after we had this chance to win another cup ... this moment is difficult."
Earlier Saturday, Frank Lampard became Chelsea's all-time leading scorer with both goals in a 2-1 victory over Aston Villa that all but ensured the London side will play in the Champions League next season.
The 34-year-old England midfielder has yet to be offered a new contract by the club, but he gave owner Roman Abramovich food for thought as he passed Bobby Tambling's longstanding mark of 202.
The win put third-placed Chelsea five points clear of Arsenal and six ahead of Tottenham, who both have a game in hand.
Villa took the lead through Christian Benteke's 19th league goal this season but the Belgian was sent off after the halftime break.
Chelsea had lost Ramires for a second booking before the break, but Lampard fired in a left-foot shot on the hour and finished from close range with nine minutes to play to take his tally to 15 this season in the EPL and 17 overall.
However, defender John Terry and midfielder Eden Hazard were taken off with injuries and are battling to be fit for Wednesday's Europa League final against Benfica.
Meanwhile, Manchester United midfielder Paul Scholes announced on Saturday that he will be retiring for the second time.
The 38-year-old quit in 2011 but was persuaded to return by Ferguson the following year.
He has not played since January, but Ferguson confirmed that the former England international will make his 498th EPL appearance on Sunday at home to Swansea.
"He will collect an unbelievable but well-deserved 11th Premier League medal," Ferguson said ahead of his own final home match at Old Trafford.