England international Hart will undergo a medical on Friday evening before putting pen to paper on a three-year deal over the weekend.
The Clarets are facing a keeper crisis following injuries to Nick Pope and Tom Heaton, and had petitioned UEFA to be allowed to sign an emergency shot-stopper ahead of their Europa League qualifier with Aberdeen on Thursday.
But they have been working on a deal to sign Hart on a permanent basis in recent days.
City have been trying to find a buyer for Hart since Pep Guardiola arrived at the club in 2016, but have had to make do with loan moves - first with Torino and then West Ham.
Neither of those clubs had been willing or able to make the deal permanent, while other clubs had proven reluctant to offer himl an escape route from the Etihad Stadium.
City had hoped to sell Hart for at least £15m ($19.7m) in the past two windows but were prepared for a cut-price move this summer as his contract has just one year remaining.
After his two ill-fated loan moves Hart is also understood to be keen for some stability regarding his career.
He missed out on a place in England's World Cup squad - ironicially to prospective new club-mate Pope - but good early-season form could see him come back into Gareth Southgate's thinking.
Presuming the move to Turf Moor is completed, Hart will leave City having made 348 appearances, winning two Premier League titles, an FA Cup and two League Cups.
He has also won the Premier League Golden Glove for the most clean sheets on four occasions and was twice named in the PFA Team of the Year.