The Colts released a statement on Jim Irsay's health ... saying he is still recovering from his "respiratory illness."

"We will have no further comment on his personal health, and we continue to ask that Jim and his family's privacy be respected."

Jim Irsay was found laying in his bed unresponsive, cold to the touch and gasping for air during a suspected overdose in December ... this according to police in Indiana.

Carmel Police Department documents, obtained by TMZ Sports, show cops were dispatched to Irsay's Carmel residence on Dec. 8 at around 4:30 AM ... after someone said they had found the Indianapolis Colts owner unconscious on a bathroom floor with a blue skin tone.

When police first arrived on the scene, they say Irsay had been moved to his bed ... where he was struggling to breathe and had a weak pulse and constricted pupils.

Police say in the docs an attempt to wake him with a sternum rub was unsuccessful. They added, though, that after administering one dosage of Narcan -- a drug commonly used to revive people in opiate overdose situations -- "he responded slightly."

Cops say they were prepared to attach an AED to Irsay -- but paramedics arrived and "took over lifesaving efforts." Irsay was eventually transported to a nearby hospital via an ambulance.

Officers noted in the documents that a caretaker provided them a list of the medications Irsay had been taking ... though they wrote in the report, "At this time, it is unknown what Mr. Irsay had ingested prior to our arrival." They classified the incident in the docs as both "overdose" and "overdose/poisoning."

Irsay had not been seen around his Colts' team in the weeks following the incident ... and on Jan. 9, the team announced he'd been dealing with a "severe respiratory illness." The Colts explained in a statement that he would miss a scheduled appearance in Los Angeles with his "Jim Irsay Band" while receiving treatment for the ailment.

"He is receiving excellent care and looks forward to returning to the stage as soon as possible," the Colts said. "We'll have no additional information at this time and we ask that you respect the privacy of Jim and his family as he recovers."

We do not know if Irsay's recent hospitalization is related to the December incident at his home.

Irsay -- who's been the owner of the Colts since 1997 -- said in an interview with Andrea Kremer back in November, he's battled addiction to alcohol and pain pills during his life, requiring at least 15 trips to rehab. He also told Kremer he previously suffered an overdose after he "was trying to detox myself."

"All of a sudden I started slurring my words and then code blue, I stopped breathing," he told Kremer of the OD. "And they revived me, and the doctor goes, 'Jim, you're one lucky man because I had signed, virtually, the death certificate.'"

We've reached out to Irsay, but no word back yet.