Il Corriere dello Sport says Sörloth has accepted an offer from the Giallorossi.
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However, Roma are yet to seal terms with Villarreal over a fee for the Norway international.
Roma are preparing a new bid of €25m plus €5m in bonuses for Sorloth.
Sörloth's buyout clause at Villarreal is said to be €38m.
His contract with the LaLiga club runs until the summer of 2028.