Despite having three years remaining on his contract, Phillips is expected to depart Anfield before the end of the summer transfer window.
In order to gain regular first-team football, the 25-year-old dropped down to the Championship for the second half of last season, helping Bournemouth earn promotion to the Premier League.
Having made 17 appearances for the Cherries, Phillips is naturally wanted by head coach Scott Parker, whose backline has been weakened by the departure of Gary Cahill.
A recent report has suggested that Bournemouth are preparing to make a formal offer for the centre-back ahead of their Premier League opener with Aston Villa on Saturday.
However, according to Goal, Fulham are ready to compete with their rivals for the signature of the highly-rated Englishman.
Head coach Marco Silva has recently added Kevin Mbabu to his defensive ranks, but he is lacking a fresh face for the middle of his backline.
The Cottagers do not have the same strength-in-depth in that area of the pitch, leading to an eagerness to get a deal over the line as soon as possible.
As it stands, however, Liverpool have allegedly not received an offer which meets their asking price, said to be in the region of £10m.
The report adds that the Merseyside giants would prefer to sell Phillips rather than allowing the player to spend another year away from the club on a temporary basis.
Although Jurgen Klopp has always spoken highly of Phillips, he is only fifth choice in the pecking order and is realistically not expected to spend at least another half-a-season kept in reserve.
Before linking up with Bournemouth in January, Phillips had been provided with just 134 minutes of first-team football, all of which came away from the Premier League.
While it is suggested that two teams from Serie A are also monitoring the situation, it appears that Fulham and Bournemouth are the two frontrunners at this stage.