The fee involved for the 25-year-old Anzhi Makachkala midfielder is believed to be a Tottenham club record of £30million, but sources close to the player say he is still considering his options.
Willian, who was also strongly linked with Liverpool, would become Spurs' third record signing this summer after deals of £17million and £26million for Paulinho and Roberto Soldado.
Tottenham could recoup all of their outlay so far if they accepted Real Madrid's world-record offer for Wales forward Gareth Bale, though the club refused to comment on reports in the Italian and Spanish media that technical director Franco Baldini was in Madrid for discussions over the transfer on Wednesday night.
Spurs could yet add Roma striker Erik Lamela to their ranks in a transfer that would take their spending during this window to almost £120million.
Ousted Anzhi Makhachkala boss Rene Meulensteen believes Spurs will have landed "a major trump card" if they can seal a deal for Willian.
"I've worked with him for a short time but I really, really like him. He will be an asset for whoever's going to get him," said Meulensteen on Sky Sports News.
"If it's Spurs that's going to get him, they'll get an excellent player. He's very versatile, he can play in different positions, he's a very creative player and an excellent professional with a very good personality.
"He's going to be a major trump card for Spurs if they can get the deal to go through.
"He can beat a player, he can score a goal. I thought he was an absolute delight to work with."