Los Blancos are already preparing a deal which may include cash plus player offer to sign Bale at the end of the season. Real left-back Fabio Coentrao has been a target for Andre Villas-Boas and could move to White Hart Lane in a swap deal.
Real want to seal the transfer in advance to avoid a repeat of the delayed summer move of Luka Modric.
Modric, 27, made a £33 million transfer move to Madrid from Spurs in August but has struggled to find a place in the staring XI.
However, the Croatia international has urged his former team-mate to join him at the Santiago Bernabeu. Bale, 23, could also replace superstar Cristiano Ronaldo who is yet to sign a contract extension with Real.
Real feel an offer of around £45 million will be enough to tempt Spurs but chairman Levy wouldn't budge on the £60m price tag for their Welsh star.
Despite signing a new four-year contract in June, Bale has told friends that Madrid would be first choice ahead of Barcelona.