Targett, 30, spent last season on loan at Middlesbrough, where he impressed enough to attract fresh interest despite Newcastle United releasing him at the end of his contract.
He is set to undergo a medical before signing a two-year deal at Hull, and will join Sergej Jakirovic's squad for their pre-season training camp in Slovenia, which begins on Monday.
The move represents a swift return to the top flight for Targett, who helped Middlesbrough reach the Championship play-off final last season, only to be beaten there by Hull themselves.
His performances for Boro, where he made 46 appearances in all competitions, were enough to convince Hull to move quickly and beat their promotion-rival suitors to his signature, offering him an immediate route back to the Premier League that neither West Ham nor Norwich could match.
Hull City remain busy elsewhere in the transfer market too. The newly promoted club are also on course to complete the signing of Sweden international left-back Elliot Stroud from Mjallby for around £3m, with the 24-year-old having chosen Hull ahead of an offer from Genoa.
Aston Villa winger Leon Bailey remains another player on Hull's radar as they continue to strengthen their squad ahead of their return to the Premier League, though Celta Vigo and Trabzonspor are also credited with an interest in the Jamaica international.
Targett previously spent three seasons at Aston Villa, making 89 appearances, before a four-year spell at Newcastle in which he struggled for regular football, featuring only sporadically before his loan move to Middlesbrough finally allowed him to rebuild his career with a full, uninterrupted season of Championship football.
Newcastle boss Eddie Howe had previously described Targett as "outstanding" when he signed him permanently following an initial loan, and the left-back is understood to have retained a good relationship with his former manager despite falling out of his plans in the north east.
Middlesbrough had themselves explored a permanent deal for Targett, but those talks stalled as the summer progressed, opening the door for Hull to step in with a firm offer of regular Premier League football.
The signing continues a busy summer of squad-building for Hull as they prepare for their first top-flight campaign since 2017. The club have already brought in goalkeeper Jack Butland, while also moving on players such as Ivor Pandur and Aidon Shehu before the end of June in order to remain compliant with financial regulations.