Watford were also reported to have had a £250,000 offer accepted, which could be near to what City are now offering. A previous bid of £200,000 from City was turned down.
Brown hopes that Prutton, a former England Under-21 midfielder, could be swayed to join City as he is friends with some of the players in the Tigers squad already, and may want to take the opportunity to return to his hometown of Hull. Unconfirmed rumours say the the 25-year-old doesn't want to return to the north though.
Leeds United and Nottingham Forest have both been linked with Prutton, who is out of contract in the summer. It is possible he won't leave St. Mary's until then, when he could have a greater choice of clubs to join on a free transfer and be able to demand a higher signing on fee.
Earlier, Brown didn't rule out moves for two players he previously managed at Derby County, Seth Johnson and ex-Tiger Adam Bolder, while Bradford have received an offer from a Championship side for winger Jermaine Johnson, who's out of contract at Valley Parade in the summer.