The Egypt striker, 24, had looked set to join Birmingham for £6million until the deal fell through, with Blues boss Steve Bruce citing the player's insistence upon different clauses being inserted to his contract.
But Black Cats manager Keane, who has struggled to acquire the quality of players he desires so far this summer, has revealed he is now pursuing the frontman.
"We have made an offer to Spurs for Mido," Keane told the Daily Express.
"He is a player I would like to speak to and to bring to this club.
"Trying to get these sort of quality players is a challenge, but all I want is the opportunity to speak to the lad. If it doesn't work out, then we will move on."