The fee is £500,000 with an extra £250k depending on a set number of appearances.
The clubs have agreed that Carlisle can have a sell on attachment to the deal whereby, they will get some money if Westwood were to be sold on.
Westwood,who has signed a three year deal, is a former Manchester City youngster. He has kept 56 clean sheets in the last three seasons and was called up for the Republic Ireland last month.
Cumbrians boss John Ward, although not happy at losing a top class keeper, wished Westwood all the best. He told his club's website: "I think it was inevitable that Keiren was going to go higher, because everyone has recognised the talent that he has. I am delighted for him on a personal level, because he was worked extremely hard, but obviously I am disappointed for us as a club, because we are losing an excellent goalkeeper."
"It goes without saying that we wish him the very best of luck in his career and I think he deserves this chance to go on up the ladder. His services for Carlisle United have been excellent, and it's no coincidence that we have also seen a lot of success for the club during his time here."
"As I say, this type of thing is inevitable, and not just at Carlisle but at many other clubs as well. "Players want to play at the highest level possible and, who knows, if we'd got ourselves in to the Championship then maybe we would have had a chance. We didn't do that, and we knew that the performances he had produced were going to attract interest. He was in the PFA team, he got international recognition, so there were always going to be people watching him. For him, as a lad, I can honestly say that I'm delighted for him, and I wish him all the best for the future."