Crookes, 22, was confirmed as Hull's fifth recruit for 2013, following the Black and Whites' acquisition of Gareth Ellis, Joe Arundel, Daniel Holdsworth and Ben Galea.
"With the way we want the team to play, we identified we were going to need speed in our side, particularly the outside backs, and Jason certainly brings that," coach Peter Gentle told www.hullfc.com.
Gentle added: "He works very hard at both ends of the field, he is quick, strong and very versatile, as he can play at wing or centre and down either side. I'm sure Jason will be a significant signing for us."
Crookes is the sixth out-of-contract player to leave the Bulls, who recently came out of administration after being taken over by local restaurant owner Omar Khan.
Tom Burgess is joining his brothers at South Sydney, Olivier Elima is returning to Catalan Dragons and Bryn Hargreaves, Ian Sibbit and Ben Jeffries are all retiring, while Craig Kopczak has terminated his contract and is set to join Huddersfield.
Crookes said: "I think every player knows what a massive club Hull is.
"Coming through the ranks as a kid, I watched the likes of Gareth Raynor and Matt Sing play for Hull and they were two of my favourite players.
"For the last three years at Bradford, we've not made the play-offs and that's been disappointing but Hull have done well this year considering the injuries they've had and I think we can really push on next year.
"It's a really good club, it's run very professionally and Peter Gentle is a coach I'm looking forward to working with. I just can't wait to get started now."
- Peter Gentle
- Gareth Ellis