Murphy signed for Rovers on a free transfer over the summer, signing a deal believed to be worth around £35,000 per week.
Yet, after a reasonably successful start under Steve Kean, the former Whites skipper has dropped out of first team contention recently and is now pondering a move behind the scenes at Craven Cottage.
The Daily Mirror are reporting that any role at Fulham could involve both coaching and PR work. The player has been taking his coaching badges.
Having just moved into a new house in the capital, there could certainly be an element of truth in the rumour and, undoubtedly, any move could benefit all parties.
Failing to establish himself at Championship level may well have convinced Murphy that his playing days are now behind him and the opportunity to go back to South West London, in any role, would appeal to him.
While his most successful days may have been at Liverpool, he exerted his influence with far greater effect at Fulham and holds the club in high regard. The fans feel the same about him.
And, with such a healthy knowledge of the game, he'd bring a lot to the table as a member of staff.
Only time will tell, though.