Brown took over at the KC Stadium in December, and guided the East Yorkshhire club to survival in the Championship.
He had been linked with a move to Newcastle as assistant to new Toon boss Sam Allardyce, with whom he worked at Bolton Wanderers.
"It is great news for us. I am delighted," Hull chairman Adam Pearson told the Hull Daily Mail.
"The contract is being drawn up and hopefully Phil should be signing on Tuesday or Wednesday."