The Irish international's contract expires in the summer and clubs in England and Holland are watching the situation with interest.
Dutch outfit AZ Alkmaar have reportedly offered the 23-year-old forward a lucrative five-year contract to lure him to Amsterdam.
AZ are being bankrolled by their wealthy chairman Dick Scheringa who has already financed the move of Ali El Khattabi from Sparta Rotterdam.
Alkmaar had a poor season in the Eredivisie finishing just above the relegation zone and Scheringa claimed that the players didn't have the right mentality.
The chairman is keen to bring in some new blood for next season and Connolly could be one of many new faces at the Amsterdam club.
Meanwhile, Alkmaar's neighbours Ajax have had another player criticise the coach Co Adriaanse in the wake of Richard Witschge's outburst.
Defender Ferdy Vierklau told a local Dutch telivison channel: "You never know where you stand with the manager. He needs to change the way he works with people."
The 28-year-old former Dutch international has accused the Ajax coach of destroying the renowned youth system at the club, claiming: "The worst thing about the manager is his attitude towards young players. They are afraid to do anything wrong and lose confidence."
FC Zwolle are set to add striker Bert Zuurmann to their squad for next season. The 28-year-old target-man currently plays with Second Division Emmen but is due to sign a three-year deal with Zwolle in the next few days.
