The club's statement comes a day after it came out Smith had essentially revealed on a podcast he wants to sign with the Roosters from 2023.
Smith also infuriated Melbourne by suggesting the club has a culture of heavy drinking.
"Recent comments made by Brandon in a podcast, and the way he expressed them, do not align with Storm's values and the way we operate as a club," the Storm said.
"The statements have deeply offended current and former players, coaches, staff and board members and the club strongly rejects the comments made about the Storm culture."
The international hooker-forward has also come under fire for swearing many times during the podcast.
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Smith issued a statement on social media.
"I want to address current media speculation going on around me at the moment, and specifically my recent comments on YKTR Sports podcast," he says.
"Over the past two months, I've read multiple headlines about me day in day out, many of which were untrue, and none that came from my mouth directly.
"I done (sic) this podcast for that very reason. To be fully transparent about the process I've been going through.
"It's honestly been really exciting throughout and I'm truly grateful to be in the position I'm in. One thing I pride myself on is being myself, and regardless of backlash I will continue to do so.
"That being said, I do acknowledge and want to apologise for how some of (my) comments have been revealed. Specifically to the Melbourne Storm, it was never my intent to disrespect or come across as ungrateful to the club that has done so much for me and my family.
"And that goes for my coaches, members, staff, and most of all, my teammates.
"I speak from the heart – for better or worse sometimes – and I'm sorry if anyone was hurt by my words.
"But at the end of the day they are my words.
"I specifically also want to address the media honing in on my comments around the Storm culture, and it being framed as a 'drinking culture' they have going on down there. Our culture is built on hard work and effort as the main priorities.
"These comments have been taken way out of context, and I emplore (sic) anyone interested in the matter to please watch the interview in its entirety before jumping to conclusions.
"This will be the last time I address the comments made on this podcast. My focus is now fully turned to the future with the Melbourne Storm and all the amazing opportunities laid before me.
"Whatever happens I will give 100 per cent. It's all I know. If I'm to be label (sic) a terrible person so be it, if I cuss to (sic) much I'm sorry but that is me and I will continue to be me."
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