It is likely Madrid will spend more than the £80 million they paid Manchester United for Cristiano Ronaldo in 2009.
Madrid coach Carlo Ancelotti told a news conference at the club's Los Angeles training camp: "I believe the club is in talks to find a solution and we will see what happens."
Bale is the PFA Player of the Year
Should Madrid trump the Ronaldo fee, they will have set the world record for most expensive player five times this century.
Luis Figo, Zinedine Zidane and Káka all previously commanded unprecedented fees.
Ancelotti also dashed any hopes Spurs fans might have of a Luka Modrić return to White Hart Lane.
Modrić has been touted as a possible makeweight in a Bale deal, but Ancelotti re-emphasised the Croat's importance to the Madrid squad.
"It is not true about Modrić - he is a player very important to us. Modrić will play with us this season," Ancelotti said.
Modrić and Bale in happier times at Tottenham
"The club is talking right now to find a solution to the case of Bale, but there is no news in this regard."
"It's hard to talk about Bale as a player of Madrid, let's see what happens. I am very happy with the squad and with the work we are doing.
"I've never talked to Bale because they usually do not talk to players from another team and now we would just talk to him.