The 25-year old signed for the Bairns in June 2019 but his time at the Falkirk Stadium was blighted by injury.
Former Falkirk co-manager Lee Miller once described the winger as one of the unluckiest players he had ever come across in terms of injury.
The former Dunfermline Athletic and Dundee United player, who had a brief spell with Rovers in 2016, made just six appearances in all competitions last season, scoring once in a 3-0 league victory over Dumbarton.
He was not one of the nine players who Falkirk confirmed would be leaving the club last month upon the expiration of their contracts but the club have made no official conformation whether a new deal was offered to Connolly.
They did however secure the services of Ben Hall and Charlie Telfer for another season when the pair signed new deals last week.