The Football Association said it was "appalled" and "dismayed" by the abuse, while Instagram has confirmed that the user in question's account has been shut down.
And midfielder Carney has now confirmed that Chelsea have reported the incident to the police.
"I would like to take a moment to address the vile messages aimed at myself and my team-mates on social media this week," Carney wrote in a statement posted on her Twitter account.
"This kind of abuse is abhorrent, totally unacceptable and very upsetting.
"It was swiftly reported by the club as is standard practice. I now trust it will be dealt with by the correct authorities to ensure this doesn't happen again either to myself or anyone else."
Chelsea men's manager Maurizio Sarri said on Friday that he had watched the Blues' 1-0 victory over Fiorentina, and the former Napoli boss was left disgusted by the comments.
"I watched the game, she's a very good midfielder, she scored," he told a news conference.
"About the situation, it's not acceptable, it's disgusting.
"I don't know what else to say. I don't use them [social media platforms], I think you can use for work, not for socialising. I have trouble speaking to my friends on the phone, I like to see them face-to-face. It's not my world."
Carney and Chelsea are next in action on Sunday, when they face Birmingham City in the Women's Super League.