The Manchester United attacker is on a season-long loan at Barca, with the deal including a €30m buyout clause.
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Mundo Deportivo says Rashford's recent form has Barca directors now leaning towards triggering his option after previous doubts.
The England international has been involved in two of Barca three goals over the past two games.
Barca coaching staff are pleased with Rashford's form and impact, though the club's top brass will hold fire until they see whether he can successfully complement a fit-again Raphinha over the long-term.