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.
Liverpool suffered their 14th defeat of a hugely disappointing Premier League campaign on Sunday, a 1-0 reverse at Swansea.
Dalglish delivered their first silverware in six years - the Carling Cup - and guided the Reds to the FA Cup final.
But the Americans are desperate to challenge for the Premier League title and play in the Champions League.
In the last month the restructuring of the club has led to the departures of director of football Damien Comolli, head of sports science Peter Brukner and, on Friday, director of communications Ian Cotton.
And newspaper reports suggested the club's bosses are lining up Wigan manager Roberto Martinez to take over from Dalglish.
Defeat at Swansea 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.
The outcome of Liverpool manager Dalglish's trip to Boston to meet the club's owners has yet to emerge but the absence of any definitive statement from Fenway Sports Group has only fed speculation about the Scot's future.
Dalglish returned home from his short visit to Boston yesterday [Tuesday], Press Association Sport understands, but in the information vacuum which followed the rumour mill went into overdrive.
Unsubstantiated claims ranged from the 61-year-old being sacked by principal owner John Henry and Tom Werner to him resigning.
With no information forthcoming from either FSG or the club Dalglish's position was left open to further debate, which had not been helped by speculative weekend reports linking Wigan boss Roberto Martinez with his job.
Just over a month ago Werner, in the wake of the departure of director of football Damien Comolli and head of sports science Peter Brukner, stated FSG had "great confidence in Kenny and he enjoys our full support".
Since then there has been no further comment from the ownership despite Dalglish winning just two of their last five league matches and losing the FA Cup final to Chelsea.
Dalglish, along with coach Steven Clarke, flew to the United States after Sunday's defeat to Swansea to present their view of what has been a disappointing season.
It was always designed to be part of the review process as Dalglish, who still has two years to run on a contract he signed just over 12 months ago, had confirmed some weeks before he would be providing a detailed report of his assessment of the last 10 months and his thoughts on plans going forward.
But a 14th loss of the league campaign at Swansea condemned Liverpool to eighth place and was their worst finish for 18 years and their lowest points tally (52) since 1953/54.
That will have left Dalglish and Clarke answering some testing questions from Henry and Werner about what progress was being made with their investment, having spent over £200million to buy the club in October 2010 and then heavily backed their manager in subsequent transfer windows.
After their meeting on Monday Dalglish made plans to return home, which partly explains why he was not pictured alongside Clarke, Henry and Werner at a baseball match between the Boston Red Sox - FSG's other marquee investment - and the Seattle Mariners.