David Moyes has now given the South Africa international, who is out of contract at the end of the season, permission to discuss a potential move in the current transfer window.
It is thought the 28-year-old has already held talks with Barclays Premier League champions Chelsea but those negotiations have broken down.
Communication breakdown: Pienaar will not be joining Chelsea when he calls time on his Everton career
Pienaar, who will be available on a free transfer in the summer, is now believed to be close to signing a pre-contract agreement with a major club on the continent.
Moyes said: 'We have had two offers and one of those offers has been accepted. We have allowed his representatives to speak to the other club, but whether it goes through or not I don't know yet.'
Pienaar has made a big impact at Everton since signing for £2million from Borussia Dortmund in 2007.
He was the club's player of the season last year but the Merseysiders have been unable to tie him to a new contract.
Talks with Pienaar had stalled but Moyes admitted that a contract offer was rejected some time ago, making his departure inevitable.
When asked if Pienaar had decided he wanted to move on, Moyes said: 'I think the player probably decided that when he turned down his contract six months or a year ago, whenever it was. But he has done very well forus on the pitch, so we have no complaints.'
Pienaar's form this season has not matched that of previous years but heremains an important member of Moyes' side.
The club hope they can now recoup a fee for him rather than seeing him run down his contract.Moyes might appreciate the money having said he has no money to buy players this month and that any new recruits will be loan signings.
Striker Yakubu became the first high-profile player to move this month, joining Leicester on loan on Thursday with a view toa permanentdeal.
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