The west London club are keen on 6ft 8in Ivory Coast striker Lacina Traore, but have been made aware that Liverpool are also keen on taking him from Cluj.
Caracciolo is reluctant to leave Brescia for anyone but his favourite team - AC Milan - but there seems little prospect of a dream move to the San Siro.
Tall order: Caracciolo (centre) could cause Premier League defenders problems
Italy's top sports paper Gazzetta dello Sport say Fulham are interested in a January move and it would be difficult for cash-strapped Brescia to resist a decent bid.
Juventus are considering him now that Manchester City have all but won the race to sign their top target Edin Dzeko from Wolfsburg.
However, the Turin giants have other names higher on their wish list and Fulham could take advantage.
Brescia admit they don't want to sell Caracciolo, who is key to them fighting relegation from Serie A this season.
The 29-year-old struck 24 goals in 35 appearances last season to inspire promotion from Serie B and is their top scorer with five in 15 appearances in this campaign.
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