Despite winning the League Cup in February and reaching the FA Cup final, the Reds' eighth-place Premier League finish was their lowest in 18 years, and their overall points total of 52 was their lowest since 1954.
The club's American owners are eager to challenge for top honours, with a Premier League title as well as a return to the Champions League being their highest priority.
Dalglish flew to the States with assistant manager Steve Clarke to discuss future plans and several reports had suggested that the Scot had decided to quit after a season-and-a-half back in charge.
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Recently, an internal re-structuring at Anfield saw the exit of director of football, Damien Comolli along with other senior staff.
Reports suggest that Wigan's Roberto Martinez is a target for the Anfield board if Dalglish does indeed leave his post.