The midfielder, 23, made 11 appearances on loan for the Clarets last season and Brian Laws turned to him again after midfielder Chris McCann was ruled out for six months with a cruciate knee ligament problem.
"We know he's a good quality player. He's hungry and had a great experience with us last year," Laws told BBC Radio Lancashire.
"It's been the main objective for us to try and get him in as soon as we possibly can, although it's been a very long process. We've had to be very patient because Jack's tasted the Premier League and we knew that if any club came in for him we'd have no chance."
Cork made 35 appearances for the Iron, scoring twice, as the club were relegated from the Championship in the 2007/08 season.
He adds to a strong former Iron contingent at Turf Moor. Manager Laws and his assistant Russ Wilcox were first paired together at Glanford Park.
Graham Alexander, who combines playing duties for the Clarets with a coaching role, began his career in North Lincolnshire and forward Martin Paterson played alongside Cork at Scunthorpe.