Gary Megson's former assistant manager performed a similar role at the City Ground last season and remains on the coaching staff under new boss Colin Calderwood.
Hull, who lost manager Peter Taylor to Crystal Palace recently, want Colchester boss Phil Parkinson as his replacement but no agreement has yet been reached.
Forest have given permission for talks to take place with Barlow.
Former boss Megson meanwhile has been linked with the vacant job at Carlisle.