According to the Sunday Mirror, Toffee's chief David Moyes - who is in South Africa as a radio pundit - has reportedly met with Pienaar at the host nation’s training camp.
A deal believed to be worth £55,000 a week is thought to have been put forward to the South Africa international, and the Merseyside club are now awaiting a response.
Everton are thought to be in a hurry to tie Pienaar down, with just one year left on his current contract and mounting interest in his services from Tottenham Hotspur.
Spurs boss Harry Redknapp has been weighing up a new £10 million bid, but it is said that Moyes is reluctant to let Pienaar leave for anything less than double that amount.
Moyes has also told the Everton board he is prepared to keep Pienaar for another year and let the South African leave in a cut-price deal next summer.
However, the latest development keeps Pienaar’s future very much up in the air and it remains to be seen whether he will be at Goodison Park next season.