The 45-year-old led the Cumbrian side to seventh in the National League this season, seven points off the play-offs.
Pools defender Matthew Bates had been in caretaker charge of the side after Dave Jones left in April.
Barrow owner Paul Casson has stated Cox has a sizeable buy-out clause.
Hartlepool's relegation on the final day of the season meant they dropped out of the Football League for the first time in 96 years.
Relegated League Two side Hartlepool United have been given permission to speak to Barrow manager Paul Cox about their vacant managerial position, reports BBC Radio Cumbria.