Anfield currently holds around 45,000 fans, significantly less than the stadiums of rivals Manchester United and Arsenal.
The club's owners want to increase the capacity to more than 60,000, but have yet to decide whether to redevelop their current ground or to move to a new stadium on nearby Stanley Park.
Principal owner John Henry, who successfully redeveloped Fenway Park as owner of the Boston Red Sox baseball team, expressed a preference to remain at Anfield on his Twitter account last week.
But in a statement on the club website, managing director Ian Ayre said logistical and geographical issues could thwart their ambitions to stay put.
"It looks increasingly unlikely there is any way we can move forward on a refurbishment of Anfield unless there are significant changes in those areas," he said.
Ayre explained that a redevelopment of their current stadium would cost significantly less while delivering roughly the same amount of revenue as a new arena.
"It's disappointing that based on where we are at the moment, we seem to be unable to press on with the more viable economic option of a refurbishment, but we remain committed to finding the best possible long-term solution," Ayre added in the statement.
The ability to find a naming rights partner could have a significant bearing on the final decision.
Ayre has previously ruled out renaming Anfield but said the club are in dialogue with several brands about the naming rights to any proposed new stadium.
Several Premier League clubs have already used their stadium names to generate revenue. Manchester City announced a hugely lucrative deal with Etihad Airways on Friday, reportedly worth 150 million pounds over 10 years.
