Byrne arrived on Wearside from Brighton & Hove Albion in 1991 but only stayed for one season at Roker Park.
It was a memorable one as Crosby guided Sunderland to safety in the old Second Division and took the club to its fourth FA Cup final against Liverpool at Wembley Stadium.
After a spirited performance, the Black Cats lost 2-0 to the Merseyside club.
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Byrne, however, had proved to be Sunderland’s cup hero that year after scoring in every round except for the final.
But after 33 league appearances and eight goals, Byrne joined Millwall following the 1992 FA Cup final after spending just one campaign in red and white.
Crosby, though, has now lifted the lid on the reasons behind the striker’s Roker Park departure.
“When we got to the final and obviously we lost he decided he wanted to leave because he wanted more money – which any player would do,” Crosby said to the FANS podcast.
“But he’d been an absolute hero and he’d done the job that was asked of him but he wanted to leave and he wanted more money.
“I tried to get him more money but he wanted too much.”
Crosby then shared an amusing anecdote regarding Byrne’s request to leave Sunderland.
“The only thing was that when he asked me for it, we were sat in a bar in Magaluf after the cup final and I was like, ‘Byrney, do not ask me about your contract while we’re having a few pints!’
"This was the first night we were there! So I said, ‘When we get back, I will discuss it with you but I’m not discussing it here.’
"I still keep in touch with Byrney he was a lovely lad but he went to Millwall and got more money.
"But as I always said to players, ‘Why would you want to leave Sunderland?’
“No disrespect to Millwall because it’s a good club, but why would you want to leave Sunderland to go there?”