Reports had suggested that Bale would finalise his £93million transfer to the Spanish capital on Friday afternoon.
But it is understood that Spurs are unwilling to sanction Bale's move until they have signed at least one new player to fill the Welshman's position.
Tottenham looked to be favourites to sign Willian, whose best position is the number 10 role Bale operated in last season, but Chelsea have now entered the race for the Anzhi Makhachkala player.
Chelsea contacted Anzhi late on Thursday evening to lodge a bid of £32million, which was accepted.
Willian was very close to agreeing personal terms with Tottenham at the time following his medical at the club, but Press Association Sport understands he will now hold talks with Chelsea on Friday afternoon to see if he prefers a move to north or west London.
Chelsea will be able to offer the Brazil international Champions League football, but he faces tough competition from Oscar, Juan Mata, Eden Hazard and Andre Schurrle for a place in the starting XI.
Even though Spurs have spent £60million this summer, Willian would appear to stand a much better chance of making Andre Villas-Boas' starting XI.
Spurs boss Villas-Boas is confident of signing Willian as he is close to the player but Chelsea hope the Russian connection between fellow oligarchs Roman Abramovich and Anzhi owner Suleiman Kerimov could swing the deal in their favour.
Tottenham will wait to see how the Willian transfer unfolds before giving the green light for Bale to join Madrid.
Spurs are also in talks with Roma officials over the transfer of 21-year-old Roma midfielder Erik Lamela.
Lamela, who can play on either wing or through the middle, wants to move to Tottenham to play in Europe following Roma's failure to qualify for the Champions League.