The Vogue singer, 66, said she had been a fan of Sir Elton since she was a teenager.
"Seeing him perform when I was in high school changed the course of my life," the post read.
"Over the decades it hurt me to know that someone I admired so much shared his dislike of me publicly as an artist. I didn't understand it."
The strain on the pair's relationship has its roots in the noughties.
In 2004, Sir Elton ridiculed Madonna's nomination in the Best Live Act category at the Q Awards, using explicit language.
He asked: "Since when has lip syncing been live?"
"Anyone who lip syncs in public on stage when you pay £75 to see them should be shot," he continued.
Madonna's team denied that she did not sing live.
Sir Elton doubled down on his criticism in the years that followed, and in 2012 said Madonna didn't stand a "chance" of beating him to the Golden Globe for best original song.
After picking up the award for her song Masterpiece, Madonna told reporters she hoped Sir Elton would "speak to me for the next couple of years".
"He's been known to get mad at me."
In her post on Monday, Madonna suggested the reconciliation may lead to a musical partnership.
"He told me [he] had written a song for me and he wanted to collaborate."