Spurs stepped up their attempts to sign the South Africa international on Monday night in response to competition from London rivals Chelsea.
Chelsea had a £3million bid accepted last week but Spurs' own initial offer, believed to be around £2million, was rejected by the Merseyside club. Everton have decided to sell the influential playmaker, who is out of contract at the end of the season, now rather than lose him on a free transfer in the summer.
Spurs are thought to have been interested for some time and were considering offering a pre-contract deal for the end of the campaign but Chelsea's intervention forced them to enter the market.
Manager Harry Redknapp said over the weekend he was unsure whether Spurs could match Chelsea's offer, but a compromise figure of £2.5million may now have been agreed.
Everton have disclosed no details of the bids but confirmed last night Pienaar had now been given permission to discuss terms with Spurs as well as Chelsea.
An article on the club's website, www.evertonfc.com, read: "Everton can confirm they have agreed terms with Tottenham over the possible transfer of Steven Pienaar.
"Confirmation of the agreement with the north London club comes just days after a bid for the South African international was accepted from Chelsea."
Pienaar joined Everton on loan from Borussia Dortmund in 2007 and the move was made permanent the following year for £2million.
He has become one of the club's most influential players and was named player of the year after an outstanding 2009-10 season.
Source: PA
Source: PA