The 15-year-old, who had trials at Manchester City and Manchester United last season, has long been tipped for the top by coaches at the League Two club.
He made his first senior appearance for the Mariners as a substitute in a pre-season friendly at Winterton in the summer, aged just 14.
Barlow agreed to leave Blundell Park over the weekend for a fee understood to be in excess of £100,000.
Grimsby chairman John Fenty said: "Clearly (Hull manager) Phil Brown is interested in the long term. He personally contacted me to ask about Jack. We then spoke chairman to chairman. When all is said and done, they have the ability to get Jack.
"Our view is, what was on the table was far, far better than if they had just taken Jack and then it would have ended up at a tribunal."