The Irishman has seen three bids for the 22-year-old rejected by his club, with manager Chris Hutchings insisting the player has told him he does not want to leave.
However Keane, who on Tuesday made his fifth summer signing when he paid Norwich £1.5million for central midfielder Dickson Etuhu, is ready to return to the JJB Stadium in a fourth attempt to prise the £5million-plus-rated England Under-21 international away.
He said: "Just because a club says a player is not for sale - if that was the case, nobody would ever move club.
"Chris Hutchings was quick to come out through the media saying that.
"Managers have rung me about players and we like to try to keep these things private.
"But some managers cannot get it out quick enough that we have made enquiries for players.
"I suppose every manager is different, but because a manager says a player is not for sale doesn't mean it is over.
"I am sure Everton told Manchester United Wayne Rooney was not for sale, but they got him, so we will see."
Keane is showing little sign of ending his recruitment drive early amid reports that the Black Cats have increased their bid for Hearts keeper Craig Gordon to a staggering £9million after having their previous approaches rebuffed.
The club also has an interest in 31-year-old defender Bobo Balde, a former Celtic team-mate of Keane's, although the Sunderland manager has played down talk that his arrival could be imminent.
He said: "There is nothing going on at the moment, I have to say. Having lost Jonny (Evans) last year, we have brought one or two in.
"We will see, we will see how that pans out in the next few days, but there is nothing concrete at the moment."
However, Keane was swift to dismiss claims from Iraq skipper Younis Mahmoud that he had turned down the chance to sign for the Wearsiders.
He said with a smile: "He is one of 400,000 who has turned us down in the last few months.
"I have never heard of the lad. We have never spoken to him - he must be in cuckoo land saying he has turned us down because I haven't got a clue who he is."
Keane has worked hard all summer in an effort to strengthen his squad, and his efforts paid dividends once again today when Nigeria-born midfielder Etuhu followed Greg Halford, Russell Anderson, Michael Chopra and Kieran Richardson to the North-East.
The 25-year-old will bring a physical presence to the middle of the field, and Keane was delighted to have landed him.
He said: "He is a big physical presence and he knows where the back of the net is.
"The most important thing I am trying to build at the moment is competition for places.
"They always talk about the top four having big squads and that's what I am trying to build here.
"Good players will push each other and he gives us something a bit different to what we have got."
Keane has now spent around £16million, but amid claims he has paid over the odds for Chopra at £5million and Richardson, who is understood to have cost around £500,000 more, he is confident he has got value for money.
He said: "It is good value for money. When you look at some of the other strikers in particular and other players, the values they are going for, I have to say I think we have got good value, there's no doubt.
"They are good ages and are very rarely injured. It's very hard to get goalscorers anyway and with Kieran, we have got an England international who is a fit, strong quick lad.
"I suppose only time will tell, but I have got great confidence they will turn out to be shrewd signings."
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