The 16-year-old striker is one of the most highly-rated players in his age group, and Manchester United scouts were impressed with his performance when Fulham under-18s visited the Manchester club earlier this month.
Both Liverpool and United are thought to be willing to pay up to £1m in compensation for Roberts, who scored the only goal of the game as his under-18s side lost to United 3-1.
The report claims the Anfield club have been long-term admirers of the England under-17s forward and were believed to be the front-runners, but United have stepped up their pursuit in the last month.
The young striker only turned 16 in February, but he's already established himself as a regular in the Fulham under-18s side.
The west London club rate him highly, and the report claims they're desperate to keep hold of him and have no desire to sell - despite interest from Liverpool and United.
Fulham are unable to tie down Roberts to a long-term professional contract because he's yet to turn 17, but an agreement between player and club over a future deal grants Fulham a certain amount of security.
The transfer of young players not subject to contracts can involve lengthy tribunals which set compensation payments between buyers and sellers.
And in this case, United and Liverpool are both thought to be prepared to offer a deal worth up to £1m including add-ons for the England youth team player.