The 66-year-old revealed that personal terms with Bent have already been agreed and all that is left is to negotiate a fee with Aston Villa.
"We were with Darren's agent for two or three hours, we thrashed out a deal," he told Sky Sports News. "We are still in negotiations with Aston Villa so there is a really strong possibility there.
"Again I hear the latest is that somebody else has all of a sudden shown interest in the player so we have competition from various clubs around the country, but I am not letting go."
Asked if Newcastle are close to signing Remy, Kinnear said: "Yeah I think very much so. I would like to think we are very, very close to agreeing [a deal].
"These are players that me and [manager] Alan [Pardew] have spoken about. These are players that we've all agreed at the club are the type of player we want to bring to Newcastle."
Newcastle are yet to sign a player this summer.