The 26-year-old will join the Reds as a free agent on July 1st after turning down a new deal with the Scottish Premiership.
Cole, son of former Manchester United and England striker, was on loan with the Manchester City Tykes in 2014-15.
“I’m really happy to be back. I was here a few years ago and really enjoyed my time,” he told the club’s website.externer-link
The US international Daryl Dike, who scored nine goals in 19 league games in the second half of the season, has now returned to the parent club Orlando City.
Boss Valerien Ismael said: “It is important that we continue to make progress as a club , and closing this deal so quickly is a clear sign that we want to be competitive next season. “