Although the fee is undisclosed, it has been reported as being more than the £1.5m up front that Celtic paid the Sky Blues last July and it is believed that Ray Ranson had a sell on clause inserted into the deal.
Fox signed a three and a half year deal at Turf Moor after playing only 22 matches for the Glasgow based club.
He won't be involved with Burnley's game against Chelsea as he is waiting for international clearance.