Spurs last week accepted a £23m bid for Steed Malbranque, Pascal Chimbonda, Younes Kaboul and Teemu Tainio and the Sunderland manager spoke with them individually at the weekend.
"If anything is going to hold things up now it is a few loose ends between Spurs and the players," said Keane, who also covets Tottenham's Darren Bent. "I was impressed with the four when I met them and I hope I made a positive impression on them. I enjoyed having the opportunity to speak to them and I think they were made some very good offers."
Although Kaboul's agent said the defender would not join Sunderland "even if there was an earthquake", the money on offer at the Stadium of Light appears to have prompted a rethink.