AS Roma are looking for a fee in the region of €40m for the 23-year-old, and neither English club has yet come close to matching that valuation in their contacts with his representatives.
The Serie A side are reluctant to offer any discount, pointing to Soule's age and the room he still has to develop as reasons to hold firm on their asking price, even amid growing interest from England. Sunderland's pursuit is understood to be led by sporting director Florent Ghisolfi, who knows the player well from his time in the same role at Roma, having originally helped bring Soule to the club from Juventus.
Sunderland's interest in a marquee attacking addition comes as the club also tidies up other business at the opposite end of their transfer plans. The club have confirmed that full-back Timothee Pembele has left permanently to join Le Havre, making his most recent loan spell in Normandy a permanent switch.
Pembele managed just eight appearances for Sunderland after signing from Paris Saint-Germain in September 2023, with recurring loan spells back at Le Havre limiting his opportunities on Wearside. He featured 35 times in Ligue 1 across those loan spells with the French club, form that ultimately convinced them to make the move permanent.
The 23-year-old, who has one senior cap for France, will now continue his career in Ligue 1 on a long-term basis having struggled to establish himself in the Premier League after arriving from the Paris giants.
Between a potentially significant incoming move for Soule and the confirmed departure of Pembele, Sunderland's summer business continues to take shape as they prepare for a first-ever Europa League campaign following a seventh-placed Premier League finish last season, with further activity expected at both ends of the squad before the transfer window closes.