The Republic of Ireland international will join Boro on an initial three-month loan with a view to a permanent switch.
Boro play Newcastle on December 19, so they are not going to finalise the move until this Friday.
Hull boss Phil Brown: "We talked long and hard over the last two weeks over who we would let go.
"We always look to make decisions with a view to progress and as a result, we've allowed Caleb to talk with Middlesbrough.
"We've always been looking for someone to finish off the work of a more and more promising squad and now more than ever, there are a lot of players on the front line competing for places.
"Caleb didn't score in his four games and didn't really look like scoring either.
"But you're always getting pictures of every individual when they come into work, not just in a game situation."