The 25-year-old made 32 appearances for Burton last season and earned his first Northern Ireland cap against New Zealand on 2 June.
"We are disappointed with his decision," said manager Nigel Clough.
"But, more than that, by the way things have been conducted by the player and agent over the last couple of months."
Flanagan joined Burton in 2015 on a free transfer from MK Dons and helped the club to promotion to the Championship in his first campaign.
"He had an offer shortly after the end of the season and it has dragged on since then," continued Clough.
"Although his agent has had some unfortunate problems to deal with, it wouldn't have gone amiss to keep us informed.
"Eventually, we had everything agreed and Tom was due to sign today (Wednesday) and we got a call when all the paperwork was done saying he wasn't signing.
"That, coming the day before pre-season starts, leaves an unpleasant taste. I had a text from Tom last week saying he wasn't interested in any other offer. What's also disappointing is that we had agreed to all his terms."
BBC Sport has contacted the Professional Footballers' Association for a response to Burton's statement.
Burton Albion have announced that Tom Flanagan is to leave the club, despite claiming the defender had "verbally agreed" a new contract.