A Wisconsin gym is catching major heat over posting a workout titled "I Can't Breathe" ... and now, the fitness center's owner tells TMZ Sports she's "deeply" sorry for the offensive sign.

Here's the deal ... Jen Dunnington, co-owner of Anytime Fitness in Wauwatosa, WI, says one of her trainers attempted to "honor" George Floyd by posting a workout for members this week.

She says the group of exercises was meant to "be a workout so hard that we felt what he felt" ... but when people in the community viewed the whiteboard for the workout, it was immediately hit with backlash.

Hundreds of people commented on the sign -- which featured the words "I Can't Breathe" and "don't you dare lay down" with a drawing of a kneeling person -- calling it inappropriate and appalling.

Dunnington tells us she regrets the sign's verbiage, saying, "I deeply apologize."

"I was not aware of the signage that it was posted in that way," she said. "I didn't read it in that way, but I do definitely apologize that those words were on there."

Dunnington -- who posted a video apology to the gym's social media page but deleted it shortly after -- insists she and the gym did not mean to offend anybody by the workout.

In fact, Dunnington says the "I Can't Breathe" line was meant to be a motivator ... calling the line a rallying cry used by protestors all over the country.

As for the "lay down" line ... Dunnington added, "I apologize that it said that, the line on there should not have been on there."

For corporate Anytime Fitness' part, it released a statement prior to Dunnington's gym's workout ... saying, in part, "We are committed to listening and learning, so that we can take action that helps our communities heal and grow in real and meaningful ways."