Spurs now hope the midfielder is granted a work permit in time for him to make his debut at Newcastle on Saturday.
They had seen an initial bid of £2m turned down before settling on a compromise figure of £2.5m with the Toffees after Chelsea had had a £3m bid accepted.
Tottenham boss Harry Redknapp, speaking after his team's 0-0 draw with Manchester United on Saturday, said: "I think he is a good player. He can play left or right or down the middle. I like him.
"I would not want to be putting the chairman under any pressure to buy him, but at £2m or £3m, he's certainly not dear is he?"
Pienaar joined Everton from Borussia Dortmund in 2007, initially on a season-long loan, and went on to make 132 appearances for the Toffees, scoring 12 goals.
He initially made his name in Europe at Ajax after joining them from their club partners Ajax Cape Town in January 2001, helping them to the Dutch title in 2002 and 2004 before joining Bundesliga outfit Dortmund in 2006.
Everton boss David Moyes had said in August that only a "sack-load of cash" would persuade him to sell a playmaker voted the club's player of the season last term.
But Pienaar, who has scored one goal so far this season, has failed to replicate that form during this campaign and Moyes has decided to cash in as his contract winds down.
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