Real allowed Mesut Ozil to join Arsenal in a €50million (£42.4m) deal following the arrival of Gareth Bale from Tottenham for a world-record fee, but they were reluctant to sell Khedira.
The German international had been heavily linked with a move to the Premier League, with Chelsea and Arsenal also believed to be pursuing the 26-year-old.
But Khedira will remain in Spain after United missed out on another midfield target in La Liga following their failure to lure Ander Herrera from Athletic Bilbao.
Speaking at press conference for the German national team, Khedira said: "The club turned down the offer and that was it.
"We didn't really have the chance to discuss it. I had a longer chat with Carlo Ancelotti a few weeks ago and it was relatively clear to me that I'd stay at Real."
Khedira forged a solid midfield partnership with Ozil during their three years together at the Bernabeu and he was sad to see his compatriot leave.
"Personally, I feel bad about Mesut's transfer," he said. "We are very, very good friends.
"His departure is a loss also from a sporting perspective. Mesut can make the difference."