Chelsea's opening bid was instantly dismissed by the Black Cats, who don't want to lose Xhaka this summer and are under no financial pressure to cash in on the 33-year-old, who still has two years to run on his contract at the Stadium of Light.
But Sunderland are also said to be aware that Xhaka is keen to reunite with his former boss Xabi Alonso at Stamford Bridge.
But Xhaka's interest in the move doesn't change the fact that Chelsea's opening offer fell way short of what the Swiss international is worth.
RECOMMENDED READING:
Sunderland paid more than double what Chelsea offered to sign Xhaka 12 months ago, and although the midfielder is now a year older - he turns 34 in September - he demonstrated last season that he's still among the Premier League's elite midfielders.
Sunderland deem Chelsea's offer to be unacceptable and have told the Blues they have no intention to let their captain leave this summer.
At Sunderland, Xhaka's worth is about more than just what he offers on the pitch as well. The former Arsenal man sets the standards at the Academy of Light on a day to day basis and his arrival last summer was transformational on Wearside.
Sunderland director of football Florent Ghisolfi said earlier this month that he's not scared of selling players, but also outlined his desire to keep key first team figures at the Stadium of Light this summer.
When news of Chelsea's interest in Xhaka emerged on Saturday, sources close to the London club claimed the Blues baulked at the prospect of paying the sort of fee that had been circulating on social media, which was between £25m and £30m.
Xhaka previously played for Alonso at Bayer Leverkusen and Chelsea's new boss pinpointed the midfielder as a key target after identifying the need for more leadership and experience in his side.
In an interview earlier this month prior to the World Cup, Xhaka told of how happy he is on Wearside.
He said: “It’s a bit like home for me.
"The city feels like Basel. Like the St. Johann district where I grew up. You can only perform at your best when you feel comfortable in a place.
“For me, the most important thing is that my family is happy. 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.”