Rashford, 28, is on a season-long loan at Camp Nou and has accumulated 21 goal contributions in LaLiga and the Champions League this season, with Barcelona now leading the Spanish top flight by 11 points with five games remaining.
Despite that, his status as a guaranteed starter under Hansi Flick is not assured, with roughly half of his appearances coming off the bench.
The Sun reports a third party is understood to be continuing discussions on behalf of Rashford, whose contract at Manchester United runs until 2027. United have made clear they are not willing to accept a discounted fee for a player they regard as having significantly increased his market value during the loan period.
Barcelona's position is complicated by their well-documented financial constraints and the broader cost of their planned summer rebuild, which includes the pursuit of Atletico Madrid striker Julián Álvarez as their primary attacking target.
They have signalled they would prefer to negotiate a second loan rather than activate the purchase option at the current price, a proposal United have so far rebuffed.
Rashford himself is understood to want to remain at Barcelona and is prepared to accept a pay cut from his £315,000-per-week United salary to make a permanent deal work. The purchase clause expires on 15 June — four days after the World Cup begins.
United have scheduled internal meetings this week to discuss outgoing players ahead of the summer window.