The Mirror says the Russian striker has demanded an exit from Spurs after finding himself relegated down the pecking order to fourth-choice striker behind Jermain Defoe, Robbie Keane and Peter Crouch.
"We had a chance to get Pavlyuchenko back last summer but it did not work out," said Spartak general director and manager Valery Karpin. "We shall see what the conditions are now.
"Due to the limit for foreign players, every team in our Premiership needs the Russians, even if they are not worth such a lot of money."
In the summer, Spartak tabled a £9million bid for Pavlyuchenko.