President Florentino Perez travelled with the team for friendlies in Los Angeles, Phoenix, Washington and Miami and expects talks to continue this week.
A Madrid source said: “Bale is the chosen man.Efforts to sign him will continue. We are hopeful a deal can be done soon.”
And now Spurs chairman Daniel Levy is set to hold showdown talks with Bale early this week.
After their £81m bid was rejected, Real will offer £51m cash plus Angel Di Maria and Fabio Coentrao, to take the deal to £95m.
Meanwhile, Spurs were last night closing in on landing Valencia’s Spanish striker Roberto Soldado for £25.8m.