Millwall - who recently announced the signing of Tigers keeper George Long - Bristol City, Cardiff and Derby County were also linked with the former West Ham defender.
Burke had an option in his Hull contract which could have been triggered for another 12 months, but national reports claimed that agreeing that option would have seen Burke's weekly salary rise over £10,000 - something the club were unwilling to sanction for the 24-year-old - out of contract at the end of June.
Former England youth international Burke - who joined the Tigers in 2018 and has since made 110 appearances for the club, scoring five goals - has now moved to Luton.
He has become the Hertfordshire club’s second signing of the summer after winger Fred Onyedinma, who joined from Wycombe Wanderers.
On Burke, who spent a loan spell earlier in his career at Bradford City, Luton manager Nathan Jones said: “We’re delighted to get this deal for Reece done because we’ve liked him for a while.
“He’s a good age, has good pedigree and is different to what we have at the club. He turned down other Championship clubs to come here, so we are convinced of his commitment to become a top player, which we feel he will achieve.
“He will give us different options which will help us to evolve, so we can’t wait to get working with him when we return for pre-season later this month.”