The 27-year-old has not played since tearing a calf muscle against Barcelona on 23 April.
Real face Juventus in the Champions League final on 3 June in Bale's home city of Cardiff.
"It's not the time to talk about the Champions League final," said Real boss Zinedine Zidane. "He's much better and has already been out training."
Bale will not be involved when Real travel to Celta Viga on Wednesday.
They will replace Barcelona at the top of La Liga with a win or a draw.
Wales forward Gareth Bale has resumed training at Real Madrid but has not trained with his team-mates.