They have not finished behind their Merseyside rivals since Everton last won the title in 1987, and the cost of surrendering bragging rights this season could be as much as £20m if it is their neighbours, rather than themselves, who get their hands on next season's Champions' League booty.
During last week, Rafael Benitez, the manager, and his captain, Steven Gerrard, insisted, in turn, that Liverpool are a better team than Everton, and as David Moyes takes his troops to Southampton today, the gap between them is reduced to four points.
Goals by Fernando Morientes, Sami Hyypia and Milan Baros ensured that Fulham have still never won at Anfield. But it was a Liverpool performance that lacked real authority, and better will be needed before Everton start to fret.
It had started full of promise. Morientes, having broken his duck with a fine strike in the midweek win at Charlton, put Liverpool in front with a superb header. At 6ft 2in, Morientes is strong in the air as well as with the ball at his feet, and he rose in the manner of the best English centre-forwards to meet Luis Garcia's cross, directing the ball across and beyond the reach of the visiting goalkeeper, Edwin van der Sar.
Morientes has needed to acclimatise to the pace of the Premiership, but yesterday he had the Anfield crowd warming to his work-rate as well as his technique. At £6.3m, the 28-year-old Spaniard is hardly a bargain, but there was evidence here that Liverpool's long court-ship of the Real Madrid striker was worthwhile. "He will be better each week," Benitez enthused. "He was five months without playing, so there will be better to come."
But the satisfaction Benitez drew from seeing a positive beginning quickly rewarded did not last long as Fulham, confident enough to attack the home side after eight games unbeaten, equalised in the 16th minute with a goal that raised more questions about Liverpool's defending.
There were plenty of red shirts covering Jerzy Dudek's goal as Luis Boa Morte swung in a cross from the left, but Andy Cole was somehow allowed to meet it with a diving header to put the London team level.
The goal, Benitez said, made Liverpool nervous. They were clearly tentative in possession, particularly in midfield, where Igor Biscan was ineffective against the tigerish energy of Lee Clark and Mark Pembridge. His replacement with Dietmar Hamann 19 minutes into the second half was overdue.
By then, Liverpool had their noses in front again as Fulham conceded a headed goal for the second time. Gerrard delivered a free-kick on the left and Hyypia glanced the ball wide of Van der Sar.
Hamann's presence brought more muscle to Liverpool's midfield but it took a third goal to bring them security. Fulham, convinced that the referee, Rob Styles, should have halted the game after Papa Bouba Diop had suffered a blow to the head in a collision with Djimi Traoré, effectively stopped playing, allowing Baros to drill home John Arne Riise's cross unchallenged.
The upshot was that Liverpool were able to play the last 15 minutes in unexpected comfort. But they will be awareit will need performances of a higher quality than this to sustain the challenge for a Champions' League place.