The Matrix is so ingrained in our minds that anyone other than Keanu Reeves as Neo is difficult to imagine, yet it turns out that was almost the case.

Producer Lorenzo di Bonaventura told The Wrap that the Warner Bros. studio wanted a big name for the lead role of computer hacker Neo — and considered Sandra Bullock (Reeves’ 1994 Speed costar) for the part in what would have been a gender-flipped film — ahead of the film’s 20th anniversary on Sunday.

“We went out to so many people I don’t remember. We were getting desperate,” di Bonaventura explained. “We went to Sandy Bullock and said ‘We’ll change Neo to a girl.’ [Producer] Joel Silver and I worked with Sandy on ‘Demolition Man’ and she was and continues to be a very good friend of mine. It was pretty simple. We sent her the script to see if she was interested in it. And if she was interested in it we would try to make the change.”

At the time, the future Oscar winner was a blockbuster regular, showcasing her versatility in films like Speed (which also co-starred Reeves). However, di Bonaventura said that discussions with Bullock were fruitless. “It just wasn’t something for her at the time.”

Bullock claimed through a spokesman that she doesn’t recall the details of the “The Matrix” proposal, but that di Bonaventura has always been supportive of her career. She did say, though, that the right person was cast in the end.

Warner Bros. wasn’t fully on board with Reeves, according to the producer.

Di Bonaventura recalls the long hunt for a big name actor who would fulfill Warner Bros.’ concept for the character before reaching out to Bullock about gender-flipping Neo. “The first movie star who says yes is Brad Pitt,” he recalls. “He’s doing ‘Seven Years in Tibet,’ and then he’s coming out of it and he’s like ‘I’m way too exhausted to take this on,’ so he’s gone.”

Next on the list was Leonardo DiCaprio. After first saying “yes,” he opted out because he didn’t want to be engaged in “another visual effects movie” after Titanic, di Bonaventurahe explained.

“Then Will Smith joins in and he drops out.”

In a YouTube video titled “Why I Turned Down The Matrix,” Smith, 50, recounted turning down the job, saying, “All right this is one of them stories I’m not proud of but it’s the truth. I did turn down Neo in The Matrix.”

Bullock stated she turned down an invitation to feature in The Matrix in 2009, although she never mentioned it was Neo.

“There haven’t been any roles that I wanted that I didn’t get, but there was a movie I wish I had done: The Matrix,” she said. “At the time [I was approached about it], it wasn’t cast with Keanu [Reeves], and I didn’t see myself with the person [Will Smith] they wanted [in it]. Later I saw the movie and loved it. It was sexy and great because of Carrie-Anne [Moss] and Keanu.”

After the picture was able to get extra funding, Warner Bros. changed their minds and cast Laurence Fishburne as Morpheus, Carrie-Anne Moss as Trinity, and Hugo Weaving as Agent Smith. I don’t think I would change anything. The bigger the star, the more likely the studio was to say yes. So we started with the very biggest and got to Keanu and he gave us the momentum,” di Bonaventura said. “The truth is, that movie rises or falls on those four.”

The Matrix became a worldwide smash after Reeves was locked in, making $463 million and resulting in two sequel films.