Chelsea wanted Rudiger to sign an extension, but sources have told ESPN the club were unwilling to meet his wage demands.
It is understood Chelsea made a final offer in excess of £220,000-a-week (€260,000) but Rudiger's deal with Madrid is thought to be worth £342,000-a-week.
Chelsea had been unable to hold further talks with Rudiger since March after owner Roman Abramovich was sanctioned by the U.K government following Russia's invasion of Ukraine.
The club were granted a special license to continue fulfilling fixtures but are prevented from signing and selling players or renegotiating existing contracts. However, Rudiger said in a piece with The Players' Tribune that Chelsea could have been more proactive in discussions earlier in the campaign.
"Unfortunately, my contract negotiations had already started to get difficult last fall," he wrote. "Business is business, but when you don't hear any news from the club from August to January, the situation becomes complicated.
"After the first offer, there was a long gap of just nothing. We're not robots, you know? You cannot wait for months with so much uncertainty about your future. Obviously, no one saw the sanctions coming, but in the end, other big clubs were showing interest, and I had to make a decision. I will leave it at that, because business aside, I have nothing bad to say about this club.
"Chelsea will always be in my heart. London will always be my home. I came here alone, and now I have a wife and two beautiful kids. I also have a new brother for life named Kova [teammate Mateo Kovacic]. I have an FA Cup, a Europa League and a Champions League medal. And of course, I have hundreds of memories that will stay with me forever."
Rudiger spoke glowingly about head coach Thomas Tuchel, however, after becoming marginalised under his predecessor Frank Lampard amid rumours he was a troublesome figure in the dressing room.
Although several teammates defended him at the time, the Germany international attracted social media criticism from some supporters, but Tuchel's arrival in February 2021 instigated a remarkable turnaround in his fortunes.
"He [Tuchel] asked about me, as a person. That was big," Rudiger said. "When Tuchel gave me a chance, I had so much motivation that I was never going back to the bench. I had made up my mind that I was going to give 200 per cent to this club, to this badge -- despite everything that was said about me. For me, after everything I endured, the Champions League was just the pineapple on top of the cake."
Rudiger's revival was completed by last season's Champions League triumph, beating Manchester City in the final.
"The match itself was beautiful, because we won against an incredible City team by defending as a unit and hitting them on the counter," Rudiger said.
"We fought for our lives, and in the end, we were champions. At the final whistle, I was running around like mad, and Tuchel just happened to be coming my way, and I gave him a big hug. That was a special moment for me, and I will always be thankful to him, because he gave me a chance when I was left for dead."