According to Tuttomercatoweb.com, the deal will reunite the siblings on the same pitch for the first time since their spell together at Adana Demirspor in Turkey during the 2021-22 season.
Barwuah, born in 1993, joins after a season in Italy's Serie D split between Vado and Desenzano, where he made 31 appearances and scored four goals.
He leaves Desenzano having just helped the club secure direct promotion to Serie C, ending his spell in Italy on a high.
Gianluca Di Marzio has reported that the move will see the forward relocate to the United Arab Emirates to link up with his older brother once again.
Al-Ittifaq are managed by president Pietro Laterza, who has been busy in the transfer market in recent days.
The club confirmed the signing of Douglas Costa on Sunday, with the Brazilian also arriving from Serie D, having spent last season with Chievo.
Costa's move brings him back together with Mario Balotelli, continuing the club's pattern of recruiting players with strong pedigree in Italian football.
Al-Ittifaq have also added Daniele Baselli to their ranks as part of the same recruitment push, further strengthening a squad stocked with familiar names from Serie A and beyond.
The club have organised their pre-season training camp in Italy, based in the Piedmont town of Bardonecchia, where the new-look squad will begin preparations together.
Barwuah's arrival completes a remarkable full-circle moment for the brothers, who first shared a dressing room in Turkey before reuniting once more in the Gulf.
Mario Balotelli, now 35, signed for Al-Ittifaq this year and has continued his career in the Middle East after his contract with Genoa was allowed to expire.
Enock's move represents a happy chapter for the family, with the brothers set to line up together once again as Al-Ittifaq continue building a squad rich in experience from Italian football.