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According to a report in the Guardian, Pienaar turned down an offer of £60 000 a week from the Toffees with the North London club set to hand the former Ajax and Borussia Dortmund player £70,000 a week over four years.
Everton boss David Moyes is thought to have resigned himself to losing Pienaar but it is unclear if the 28-year-old will join Spurs during the January transfer window or at the end of the season when his contract expires.
If Spurs insist on taking him now then they would only have to fork out a fee of £2m for his services.
With Moyes desperate to sign a striker to end his side's goalscoring woes, they might opt to cash in on their midfield star before he becomes available for free.
The Everton boss needs to generate funds if he is to make any new signings with a number of loan deals on the cards.
Monaco's Dieumerci Mbokani has been linked to the club with under-performing striker Yakubu Ayegbeni set to join Leicester City.
