According to The Athletic, Chelsea's bid has been described by Sunderland as "unacceptable," with the club unwilling to entertain any approach for a player who was instrumental in their seventh-place Premier League finish and European qualification last season.
The situation marks a sharp reversal from comments Xhaka made as recently as last month, when the 33-year-old insisted he was happy on Wearside and not considering a transfer.
"For me, the most important thing is that my family is happy," Xhaka told Blick. "And we are in Sunderland. I'm not even thinking about a transfer.
"I'm really happy to finally have a summer without any speculation about my future. That's rarely been the case in recent years. This allows me to fully focus on the World Cup."
Chelsea manager Xabi Alonso is keen to reunite with Xhaka, who played a key role in the Bundesliga title-winning season at Bayer Leverkusen in 2024. Xhaka joined Sunderland last summer following their return to the Premier League in a deal worth £17.5m.
Separately, Sunderland confirmed the departure of head of professional development Graeme Murty, who left after four years with the club's academy.
"It has been an absolute privilege to work with so many talented players, coaches and dedicated support staff during my time in the Academy," Murty said. "I'm so proud of everything we have collectively achieved over the last four years. Watching Tommy Watson clinch the Club's return to the Premier League was a moment every Academy staff member will cherish. I wish the Club continued and sustained success in the future."
The club also confirmed that academy defender Zak Johnson has joined newly promoted League Two side York City on a permanent deal, having spent time on loan there last season where he made 25 appearances. Johnson made his Sunderland first-team debut at 16, at the time becoming the youngest outfield player at the club in 50 years.