Dalglish and assistant Steve Clarke are known to have flown to the United States for talks with the club's owner John Henry and chairman Tom Werner.
And numerous reports on Tuesday afternoon suggested Dalglish has left the club.
The Americans are desperate to challenge for the Premier League title and play in the Champions League, and that was never really on the cards this season.
We reported on this website on Sunday: "Swansea will be Dalglish's swan song", and defeat at the Liberty Stadium meant Liverpool recorded their lowest total of top-flight league victories (14) and lowest points tally (52) since 1953-54.
They also scored the fewest league goals (47) for 20 years and an eighth-placed finish was their worst for 18 years.