The 19-year-old almost became a Liverpool player on deadline day.
But the two clubs ran out of time to complete a deal after agreeing on a £8m deal.
Sky Sports says the Reds believe they can secure Carvalho on a free transfer once his contract expires at the end of this season.
Fulham will be entitled to compensation for the England youth International.
Carvalho has established himself in the Fulham first-team this campaign with seven goals and four assists in 18 Championship appearances.
Chelsea, West Ham United and Real Madrid have all been linked with the Portugal native in the past.
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