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Rodgers steered Swansea's players to an 11th-placed finish, just five points behind eighth-placed Liverpool.
Liverpool have been given permission by Wigan to speak to manager Roberto Martinez about succeeding the fired Kenny Dalglish.
Andre-Villas Boas, Borussia Dortmund coach Juergen Klopp and Wigan Athletic's Roberto Martinez are among the early favourites to succeed Dalglish.
Liverpool ended the season with their lowest-ever Premier League points tally.
Dalglish led the club to success in the League Cup this season, but Liverpool finished a disappointing eighth in the Premier League.
Liverpool's emphatic 4-1 win put the Blues out of contention for an automatic place in next year's Champions League.
After Ramires put Chelsea in front after 11 minutes, Drogba scored his eighth goal in eight games at Wembley in the 52nd.
Drogba has scored in every competitive match he has played at Wembley - including FA Cup final wins in 2007, 2009 and 2010.
Liverpool have no chance of qualifying for the Champions League and are languishing in eighth place in the Premier League.
Fulham collected their first-ever win on Merseyside, after defeating Liverpool 1-0 thanks to a Martin Skrtel own goal.
Dalglish's men remain on eighth but close the gap behind Everton to two points.