Kluivert could cost around £8million and Fulham have moved for the 23-year-old as they bid to bolster their forward options ahead of their return to the Premier League.
Reports in Italy claim Roma general manager Tiago Pinto has travelled to London to secure a deal for Kluivert, a Holland international who is the son of Dutch legend Patrick.
Kluivert spent last season on loan at Nice, where he scored six goals in 29 appearances.
Duffy is set to move to Craven Cottage on a season-long loan deal after being told by Brighton manager Graham Potter that he is not part of his plans.
Fulham remain in talks over a move for West Ham centre-back Issa Diop, who is their top defensive target. They have also been linked with Eric Bailly from Manchester United and Jannik Vestergaard from Leicester.
Whites manager Marco Silva wants new signings after admitting last weekend that his side are not ready for their opening game of the season against Liverpool tomorrow.
Fulham are also in talks to sign versatile Bayern Munich forward Joshua Zirkzee.
Bournemouth and Southampton have also shown an interest in the Holland Under-21 international but he favours a move to Craven Cottage.
Fulham have made five signings this summer - Bernd Leno, Joao Palhinha, Andreas Pereira, Manor Solomon and Kevin Mbabu.