The 17-year-old moved to Wanderers on transfer deadline day after the north-west club agreed to pay a reported £1million fee for the player.
Wolves allege that Connolly, capped by the Republic of Ireland at youth level, was subject to "unapproved communication" from Bolton assistant manager Chris Evans - a former academy chief at Wolves - prior to the deal being concluded and have now reported their concerns to the authorities.
Wolves chief executive Jez Moxey said: "We have lodged an official letter of complaint, supported by considerable amounts of documentary evidence, with both the Premier League and the FA.
"This complaint relates to Bolton Wanderers and their assistant manager Chris Evans, a former Wolves employee. We allege that there has been a breach of the Premier League and FA rules in place to protect a club's academy programme, maintain player contract stability and prevent illegal approaches of players.
"It is not an issue about money or the level of compensation we received for a player we did not want to leave; it is about the protection of the club's academy programme and its development of young players, something which is of paramount importance to us at Wolves.
"We cannot sit on our hands and allow the events surrounding Mark Connolly's departure to pass unchallenged or unnoticed."