Supporters' group Spirit of Shankly have called on Liverpool chairman Tom Werner to be held accountable for the way the club is being run.
Disquiet has been growing among fans regarding the lack of a decision on whether to build a new stadium or renovate Anfield.
The lack of a prominent figure from owners Fenway Sports Group based on Merseyside, the sacking of iconic figure Kenny Dalglish as manager in May and, most recently, the apparent mistakes made in the transfer market have led to more murmurings of discontent.
Principal owner John Henry wrote an open letter to fans on Monday admitting to mistakes in their near two-year reign and outlining FSG's blueprint for the future but SoS believe more questions need answering.
"There comes a point at any football club where the chairman has to step forward on behalf of the board and be held accountable for decisions that they have made and strategies that they have implemented," said a statement from SoS, who were at the forefront of a campaign to oust former owners Tom Hicks and George Gillett.
"With key questions continuing to go unanswered, Spirit of Shankly suggests it is time for Tom Werner to be held accountable as chairman of Liverpool Football Club.
"After all, the buck stops with him, not with those acting with his authority and not with people no longer at the club."
Friday's transfer deadline day passed without a replacement being found for striker Andy Carroll, who had been loaned to West Ham the previous day.
Rodgers had wanted to bring in Fulham 29-year-old Clint Dempsey but FSG were reportedly unwilling to meet the Cottagers' inflated valuation, leaving the Reds boss with only two recognised forwards until at least January.
In his letter Henry insisted the club's ambitions were not about "cementing a mid-table place with expensive, short-term quick fixes that will only contribute for a couple of years".
That led to suggestions - dismissed by Rodgers - there was a difference of opinion between the owners and the manager.
SoS believes greater backing should have been given to the Northern Irishman just three months into the job.
"Following yet another summer where off-the-pitch activities at Liverpool Football Club have dominated the headlines, we once again find the club ownership attempting to explain away business decisions that have overshadowed footballing matters," said the statement.
"Spirit of Shankly reiterates its stance that the position of manager at Liverpool Football Club should be supported with all of the resources at the club's disposal.
"However, events of last Friday and the subsequent open letter from the club's principal owner indicate that almost two years into FSG's ownership of the club, this is not yet the case.
"There remains no ownership presence on this side of the Atlantic. It is our opinion that this situation has led directly to the 'mistakes' alluded to in Mr Henry's open letter.
"Should the club's absentee owners not wish to establish a full-time base in Liverpool it remains imperative that they appoint a chief executive of a calibre commensurate with the club's global status to act with the full authority of the owners in their absence.
"Without this it is far from sufficient for a club of Liverpool's stature to have a part-time chairman, based on another continent, with various other interests, and from whom little or nothing of consequence is seen or heard.
"In case FSG need reminding, they are now employing their third manager, have already dispensed with the services of their director of football and now accusatory fingers are pointing in the direction of their managing director (Ian Ayre)."
Liverpool declined to offer a response to the statement.