It is understood progress has been made with both Pascal Chimbonda and Teemu Tainio, while Younes Kaboul's position appears to have changed markedly since his agent Rudy Raba flatly ruled out a move to Wearside last week.
He told The Journal: "We have got an agreement with Spurs and a few of the players.
"We have virtually agreed personal terms with two or three of them, if not four. There are just a few loose ends to be tied up.
"I have been in the game long enough to know that these things can drag out a little bit, but in defence of (chief executive) Peter Walker and (chairman) Niall Quinn, from our point of view, it's been full steam ahead.
"I have spoken to Peter and progress has been made, but (with) the amount of players we are trying to sign from one club, there are going to be stumbling blocks."
Keane also admitted talks are ongoing over other targets, although he refused to be drawn on their identities.
He has been repeatedly linked with Reading midfielder Stephen Hunt, while he has also been credited with an interest in both Spurs striker Darren Bent and Everton forward Andrew Johnson throughout the summer.
Keane said: "We are in discussions regarding fees for one or two other players and that will be the case over the next two or three days.
"I would hope the Tottenham deals would be done by the weekend, and with the other couple of deals I think we might have to be patient, possibly next week."