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Former heavyweight world champion Haye, 36, made a series of graphic descriptions of hurting Bellew, with the BBBofC warning both fighters.
Bellew, 34, stopped Haye in round 11, with the Londoner subsequently having surgery on a ruptured Achilles tendon.
"Some of the comments went too far," Haye told BBC Sport.
It is expected that any sanctions against Haye will be announced after a BBBofC meeting on Wednesday.
"If I have to pay a fine, I'll happily pay and take whatever punishment I need to," he said.
Haye added he now has a "different kind of respect" for WBC cruiserweight champion Bellew and would not repeat such comments before a rematch, which he reiterated his desire to pursue.
"When you get into that beast mode before the fight, you're not thinking about political correctness, you're not thinking about fines, you're just thinking about getting in the ring," he said.
"When you're in that beast mode you say things which rub the average person up the wrong way."
Haye suffered an Achilles injury in the sixth round at the O2 Arena, with Bellew capitalising to score a knockdown.
A second knockdown in the 11th round saw Haye's corner throw in the towel, which the fighter now believes could have kept him from doing "irreparable damage" to his ankle and foot.
"It was a strange situation - I was getting into my flow and it felt like I stepped on a bear trap," he said.
"I saw Tony look down and think 'I've got you here,' so it just became about survival - trying not to get knocked out while trying to knock him out on one leg, which is pretty tough.
"I knew it was completely ruptured - the pain was terrible, when I put my foot down I could hear it crunching, but the adrenaline is so high when you're fighting you can conjure up the energy from somewhere to keep going.
"I believed if I landed that one shot I could change the outcome of the fight but I was fighting a current world champion - I gave it my best shot but came up short."
Bellew said he is considering retirement following his win, but admitted an offer for one further fight could be too lucrative to turn down.
Haye admits a rematch "solely depends on Tony" but sees a second bout as the first step as he seeks to regain a world heavyweight title belt.
"I still believe it, more now than ever, that if I'm fit and healthy, I can beat anybody," he said.
"Maybe [Tony] is happy with the victory over me and if that's the case, I wish him nothing but luck. If he wants to do it again, then great.
"Very few people on the planet believe I'm the best so I need to win that fight again, bare minimum, and then people can maybe look at me for world title fights.
"For me, focus on the leg, and I will become the world heavyweight champion."
David Haye expects a fine from the British Boxing Board of Control for his comments in the build-up to Saturday's defeat by Tony Bellew.