Madonna has said she has “buried the hatchet” with Sir Elton John and hinted she will collaborate with him, after watching the pianist and singer perform with Brandi Carlile on Saturday Night Live (SNL).

The strained relationship dates back to 2002, when John was quoted by CBS News as describing her theme to Die Another Day as “the worst Bond tune ever”.

He went on to say “Madonna, best live act? Fuck off” on stage at the Q Awards in 2004, and accused her of lip-syncing.

The pair have traded a number of remarks since, but in a post on Instagram on Monday, Madonna, whose full name is Madonna Ciccone, said: “We finally buried the hatchet.”

The singer said John’s recent appearance on SNL reminded her of when she snuck out of her home while at high school to see him perform in Detroit, which she described as an “unforgettable” performance that changed the course of her life.

She said: “I had always felt like an outsider growing up and watching him on stage helped me to understand that it was OK to be different, to stand out, to take the road less travelled. In fact, it was essential.

“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. I was told Elton John was the musical guest on SNL and I decided to go.”

Madonna said that she confronted John backstage, where he asked her to forgive him. She said the pair then hugged and the pianist told her he had written a song for her and wanted to collaborate.

John and Carlile performed Little Richard’s Bible on SNL from their collaborative album Who Believes In Angels?, which is on course to become the Briton’s 10th UK No 1 album.

In response to Madonna, John thanked her for coming to see him on SNL and for forgiving his “big mouth”.

He wrote on Instagram that he was not proud of what he had said, praising her advocacy for HIV/Aids in the 1980s and her “groundbreaking work” that paved the way for an entire generation of female artists.

He said: “I’m grateful we can move forward. I’m increasingly distressed by all the divisiveness in our world at the moment.”

John said he and Madonna had been wholeheartedly accepted and embraced by communities who are under threat around the world, and that he hoped to make “great things happen” together for those who needed support.