The as-yet unnamed agency is set to open its doors in the spring and will offer media and crisis comms training.
It is not clear what McCarthy's plans are in regard to McCarthy Communications, which provides the same services.
Lyons has played a key role in building Red’s corporate practice over his four years there, according to one source close to the agency.
"He was a significant hire and well liked, and he brought in a lot of business," the source added.
Red’s CEO Mike Morgan said: "Mark has led significant growth in our corporate practice and not only do we wish him luck in his new venture, but we will continue to use his first class media training expertise as a consultant for the business."
Lyons was a producer at ITN during the 1990s when he first worked with McCarthy, who was a reporter at the time.
He then became political editor for Granada TV and first went into PR at Lexington Communications in 2002.
McCarthy could not be reached for comment at the time of publication.
