Mitch McConnell Continues to Improve in Hospital Amid Death Rumors

Mitch McConnell is still kicking ... and despite a wave of online rumors claiming he's kicked the bucket, the longtime Kentucky Senator's team says he's continuing to get better during his recent hospitalization.

A spokesperson for McConnell tells TMZ ... "Senator McConnell appreciates the outpouring of support he's receiving while he continues his recovery in the hospital. The Senator continues to improve, and is working closely with his staff on Kentucky and Senate matters while the Senate is out of session."

The update comes after days of rampant social media speculation falsely claiming the 84-year-old Republican had died ... rumors that spread after TMZ reported new details surrounding the medical emergency that landed him in the hospital last month.

As we previously reported, McConnell was hospitalized on June 14 after suffering what was initially described only as an unspecified medical issue. At the time, a rep for McConnell said he was receiving "excellent care" ... but didn't disclose any details on the medical issue.

Then this week, we revealed 911 audio indicating McConnell was found unconscious at his D.C. home before being rushed to the hospital.

McConnell's dealt with a series of health problems over the years ... including back in February when he landed in a hospital bed due to flu-like symptoms.