The talks had already been in progress, but the situation has now become more urgent for the Cottagers.
According to La Stampa newspaper, Fulham are accelerating the progress in order to get 23-year-old Kluivert as soon as possible.
This is a deal expected to be a loan with obligation to buy for circa €9.5m.
Solomon could well require knee surgery and be ruled out for as much as three months.
He only joined Fulham last month on loan from Shakhtar Donetsk and made his debut off the bench in a 2-2 draw with Liverpool.
Kluivert was purchased from Ajax for €17.25m in the summer of 2018 and failed to make an impact at the Stadio Olimpico.
He went out on loan experiences at RB Leipzig and OGC Nice, but they were not prepared to take up the option to buy.