McCourt is a free agent after being released by Barnsley, the club he joined last summer following his departure from Glasgow.
David Flitcroft, the Bury manager who played alongside McCourt at Rochdale and signed him for Barnsley last summer, hopes the pair can renew their long-standing links. McCourt also has options in Ireland.
"Paddy's had some good offers back home and abroad," said Flitcroft. "It's a possibility still. I know Paddy probably better than anyone in terms of football and his heart might rule his head on this one or his head might rule his heart. I'm not sure which way it will go.
"He does enjoy the English challenge and it's a tough decision."