The Cottagers are preparing for life back in the Championship following their relegation from the Premier League and manager Scott Parker is eager to strengthen his attacking options.
With Ryan Babel and Andre Schurrle both leaving the club this summer and Aleksandar Mitrovic tipped for an exit, bringing in a new forward appears to be imperative for the west London outfit.
It is claimed Fulham are looking into the possibility of signing DR Congo international Assombalonga, who has finished as Boro's top goalscorer over the past two seasons.
The 26-year-old has netted 29 goals from 88 Championship appearances since arriving at the Riverside from Nottingham Forest and he is attracting interest from a number of clubs.
Swansea City and Crystal Palace are both said to be weighing up offers, which will need to be around £12miillion to land the player.