Sky Sports revealed earlier this week Real have offered a world-record fee of more than £85m for Bale who, according to Sky sources, reported for training on Wednesday and told Andre Villas-Boas he wants to leave.
And Real boss Carlo Ancelotti, addressing a news conference from his club's tour of the United States, has confirmed for the first time that talks over a transfer are under way.
"It is difficult to talk about Bale because at the moment he is not a Real Madrid player," Ancelotti said.
"I believe the club is in talks to find a solution and we will see what happens.
"I am not used to talking about players who are not in my team and it would not be right to do so."
Should Spurs agree to the deal it would represent a world record fee and eclipse Real's own record, which they hold for the transfer of Cristiano Ronaldo from Manchester United in 2009 when they paid £80million.
Former Spurs star Luka Modric has been linked with a move back to White Hart Lane as part of the deal, but Ancelotti says this is not true.
"All the players we have now are happy, we have not talked about any player leaving," he insisted.
"Modric is a very important player for us and will play with us next season."
Real players, Kaka, Iker Casillas and Sergio Ramos have already spoken out in favour of the club's pursuit of the Wales international.
Brazilian playmaker Kaka has suggested that Bale would fit in well in the Spanish capital as Real try to secure the club's first Champions League crown since 2002.
"He's a very, very good player," said Kaka on Sky Sports News. "If he comes to Madrid he will be welcomed."
"He's an amazing player with lots of technical ability, skills and tricks. We will be very happy if he comes."
Meanwhile, Swansea boss, and former Real player, Michael Laudrup believes Bale will be worth any sum his old club are prepared to pay to prise the winger away from White Hart Lane.
"Is he worth £100m? Who can tell? Who can tell what something is worth? Everything is worth what one club, one person, one company, is willing to pay. It's like buying a house," said Laudrup.
"We talk about it like merchandise and I don't like it, but it's fact that if one person is willing to pay so much for one player then he is worth it."
Laudrup also admits he can understand why Bale's head would be turned by the chance to play for one of the biggest clubs in the world.
"Real are a massive club. I don't know if he will go as I only know what I've read in the papers here and in Spain because there's huge interest in him," he said.
"There are three or four clubs who say they are the biggest in world but Real Madrid are definitely one of them.
"Tottenham are a big club but Real Madrid is a step more in every aspect, they have won the Champions League nine times and I don't know how many league titles. There is a lot of pressure.
"You have to have quality as a player but you also have to cope with that pressure and not everyone can do that.
"He has technical ability to play there but let's see what happens. I think he's a great player and he's already shown that with a fantastic season for Tottenham last season.
"But we are reading about some big numbers and if they are true maybe in the end he will go."
He added: "Big players want to play at the highest level, at a club who try and win the Champions League, who get to finals and semi-finals.
"There are six, eight, 10 clubs with that potential and Real Madrid is obviously one of them. All big players want to play for that possibility because it's special."