The former Destiny's Child singer, who sang her solo and group hits and covers at the music festival last weekend, was turned down by musician Lennox because she had a back injury and could not perform on stage.
'Walking on Broken Glass' singer Lennox told Capital FM reporter Kevin Hughes about the invitation.
Beyoncé covered Kings of Leon, Prince and Alanis Morissette during a Pyramid Stage performance at Glastonbury. After the set, she said: "This night was a dream. I felt like a rock star, and the crowd, they looked like rock stars, and I'm just so honoured, really. This is... the highlight of my career."
In December last year, Lennox named Beyoncé as one of her favourite singers, together with Shakira, Whitney Houston and Aretha Franklin.
Watch Beyoncé's new 'Run the World (Girls)' video below: