Poyet was suspended by the Seagulls following their play-off semi-final defeat to Crystal Palace, and he was subsequently sacked for gross misconduct in June.
But Bloom has revealed he received a call from the Uruguayan - who was under contract until 2016 - in March, shortly before Poyet turned down an approach to replace Brian McDermott at Reading.
At a Brighton fans forum on Thursday, Bloom said: "It got triggered by a phone call four days before the game against Crystal Palace in March.
"He phoned me up and he made it clear to me that he wanted to leave at the end of the season.
"He didn't want to discuss it at all. He said that if he could leave, if it would be allowable for him to leave the next day that would be fine.
"So this was hugely shocking to me and it was something I had to manage between then and the end of the season.
"Obviously the key was the players and the team and our promotion push. I didn't want anything to get in the way of that.
"And then lots of things happened between then and the end of the season which I don't want to discuss.
"I had a great relationship with Gus for three years. So it's just very, very disappointing the way things finished."
But Bloom is confident that the team can challenge for promotion again this season under new head coach Oscar Garcia.
"I just want to look forward now because we've got Oscar here, we've got a great coaching team and I'm looking forward to the season ahead," Bloom added.
"I don't want to put Oscar under too much pressure before the season even starts. It's always a difficult one.
"It's a very competitive division as we know and every season, at the start of the season, we have to try and have a squad we think is capable of promotion. I certainly think this season we have that."