The Dutchman will take no part in Tottenham's final pre-season friendly against Athletic Bilbao on Saturday, having picked up an ankle injury in a 3-2 win over Brighton.
Spurs open their competitive season at home to the Toffees, but their Europa League play-off tie with Hearts on August 18 may prove a more realistic date for Van der Vaart's return.
"We'll keep our fingers crossed for next week," Kevin Bond, Tottenham's assistant manager, told the Daily Telegraph.
Tottenham's first-team coach Joe Jordan, who managed Hearts from 1990 to 1993, admits both sides have a lot of foreign influence.
He does not believe that takes away from the inevitable 'battle of Britain' tag, though, and insists Spurs will take the game as seriously as any other European tie.
"Hearts have a good contingent of foreign players and so do we," he said.
"So that aspect will be with the supporters, that's how they will look at it, they will enjoy that and it will give the tie some added spice.
"But the players have played all over the world and it will be all about winning the tie and going through to the next round."
He added: "You can't predict what is going to happen but certainly we will play the strongest team that we can.
"We played over 10 months to get into the competition."
Source: Team Talk
Source: Team Talk