The 21-year-old spent the second half of last season on loan at Sevilla, but the La Liga outfit turned down the chance to sign him on a permanent basis.
Rangers are among the clubs to be linked with the Tunisia international this summer, but his move to Championship Burnley has now been confirmed.
"Hannibal has completed a permanent transfer from Manchester United to Championship side Burnley," read a statement on the Premier League club's official website.
"After spending pre-season and the start of 2024/25 as part of Erik ten Hag's squad, he now joins Burnley, who are looking to make an instant return to the Premier League under new manager Scott Parker, following their relegation last term. We wish you the best of luck with your next steps, Hannibal!"
Hannibal completes move to Championship side Burnley
"I'm very excited. Very happy to be here and I can't wait to see the fans. I have played against Burnley before at Turf Moor and the atmosphere was amazing," Hannibal told Burnley's official website.
"After speaking to the coach [Scott Parker] the project sounded good to me and hopefully we can make success."
Hannibal made the move to Old Trafford from Monaco in 2019, and he has left the 20-time English champions with a record of one goal in 13 appearances in all competitions.
The midfielder, who is believed to have cost Burnley in the region of £9m, only made six appearances for Sevilla in the second half of last season, but he will now be looking to kickstart his career in the second tier of English football.
Man United are 'closing in' on Ugarte deal
Man United are seemingly close to making their fifth signing of the summer market, with Manuel Ugarte allegedly currently undergoing his medical with the club after flying into Manchester on Tuesday night.
Ugarte is set to sign a five-year contract, with the Uruguay international preparing to join from Paris Saint-Germain.
Hannibal is expected to be one of two midfield exits for Man United in the latter stages of the transfer window, as Scotland international Scott McTominay is also set to make the move to Napoli.