The 34-year-old former Chelsea and Hibernian shot-stopper joined the Tykes in the summer of 2004 and has gone on to make over 100 first-team appearances.
But the former Republic of Ireland international has not made an appearance since the opening day of the season and has asked to leave Oakwell.
Colgan has been forced to play second fiddle to Heinz Muller since conceding four goals at home to Coventry City on August 11.
"Nick has requested to leave and we have accepted his request," manager Simon Davey explained.