Spurs were originally told the 27-year-old would cost upwards of £20million and no fee was agreed between the clubs.
"There are only small differences left," Zenit sporting director Konstantin Sarsaniya told Russian newspaper Sovetsky Sport, who suggested a deal could be concluded over the weekend.
"Zenit have decided to reduce the asking price for Andrei, but exactly how much I cannot say because the negotiations were led by Zenit president Alexander Dyukov."