The contract refusal has forced manager David Moyes to concede that he does not expect Pienaar to remain at Goodison Park once his current contract expires at the end of the season.
The 28-year-old has already rejected an improved contract earlier this year and negotiations have been ongoing since then to try and tie down the midfielder to the club.
However, Moyes told the Guardian: "I have probably resigned myself to him leaving but I knew that when there was no contract signed in the summer.
"We brought him in for £2.5m and you tend to find people in the last year of their contract have to play out of their skin to impress suitors. If it is the money they want they are going to have to play well to show they are worth the wages they want. I will try and use it to my advantage.
"There is a really good contract offer there for him but, to be fair, he has stood up for us this season and has done some really good things on the pitch."
The South African midfielder has been linked with a move away from the club since his first contract refusal earlier this year, with Premier League sides Tottenham and Manchester United being linked with the midfielder, along with Serie A and Champions League winners Inter.