Michele, 33, says in a statement exclusively shared with PEOPLE that while she does not remember making the specific remarks that Ware says she made, that is "not really the point."
"What matters is that I clearly acted in ways which hurt other people," Michele says, explaining the importance of listening and learning from those who she might have hurt in the past.
"One of the most important lessons of the last few weeks is that we need to take the time to listen and learn about other people's perspectives and any role we have played or anything we can do to help address the injustices that they face," Michele says.
"When I tweeted the other day, it was meant to be a show of support for our friends and neighbors and communities of color during this really difficult time," she says, referring to her May 29 tweet in support of George Floyd and the Black Lives Matter movement.
She continues: "but the responses I received to what I posted have made me also focus specifically on how my own behavior towards fellow cast members was perceived by them."
"While I don’t remember ever making this specific statement and I have never judged others by their background or color of their skin, that’s not really the point," Michele's statement continues. "What matters is that I clearly acted in ways which hurt other people."
"Whether it was my privileged position and perspective that caused me to be perceived as insensitive or inappropriate at times or whether it was just my immaturity and me just being unnecessarily difficult, I apologize for my behavior and for any pain which I have caused. We all can grow and change and I have definitely used these past several months to reflect on my own shortcomings."
"I am a couple of months from becoming a mother and I know I need to keep working to better myself and take responsibility for my actions, so that I can be a real role model for my child and so I can pass along my lessons and mistakes, so that they can learn from me," she concludes. "I listened to these criticisms and I am learning and while I am very sorry, I will be better in the future from this experience."
Ware, who played Jane Hayward on the musical TV series for 11 episodes in 2015 opposite Michele, responded to the Scream Queens star's tweet about Floyd on Monday, calling her out for making Ware's time on set "a living hell."
"LMAO REMEMBER WHEN YOU MADE MY FIRST TELEVISION GIG A LIVING HELL?!?! CAUSE ILL NEVER FORGET," Ware tweeted.
"I BELIEVE YOU TOLD EVERYONE THAT IF TOU HAD THE OPPORTUNITY YOU WOULD 'S— IN MY WIG!' AMONGST OTHER TRAUMATIC MICROAGRESSIONS THAT MADE ME QUESTION A CAREER IN HOLLYWOOD..." added Ware, 28.
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A source close to Michele told PEOPLE on Wednesday that the actress "has reached out to a few of her former cast members and spoke to a couple of them."
After the Ware's allegation, the meal kit company HelloFresh announced that it would be ending its partnership with Michele.
"HelloFresh does not condone racism nor discrimination of any kind," the company said in a statement on Twitter Tuesday. "We are disheartened and disappointed to learn of the recent claims concerning Lea Michele. We take this very seriously, and have ended our partnership with Lea Michele, effective immediately."
Following the announcement, HelloFresh confirmed to PEOPLE that its partnership with Michele was over. A source close to the actress tells PEOPLE that "the partnership was nearing its conclusion regardless."
Ware did not provide further detail and did not specifically accuse Michele of any racially discriminatory behavior.
Just days after the pregnancy reveal on April 27, Michele shared a sweet photo of her baby bump on Instagram.
In the image, Michele sweetly cradled her belly in her hands while wearing a long blue dress. "So grateful," she wrote alongside the image.