Bale almost joined Chinese Super League side Jiangsu Suning earlier in the summer after Real boss Zinedine Zidane said it would be best for him to leave the club, only for the Spanish club to pull out of the deal.
Per Goal's Chris Burton, Giggs said:
"I kept in touch with him, not every day but just to see if he was all right. It's obviously positive at the moment.
"He played at the weekend and did really well, and it looks like he's staying now. I've always said he's at a fantastic club but probably a couple of weeks ago we wouldn't have seen this situation.
"It was a strange situation but hopefully now it's sorted out, Gareth stays and he plays games."
Having looked as if he had little future at the Santiago Bernabeu, he started Real's 3-1 win over Celta Vigo last Saturday and capped an impressive performance with an assist for Karim Benzema.
Managing Madrid's Kiyan Sobhani and football journalist Ben Hayward believe he could still be an important player for Los Blancos:
After the match, Zidane gave a strong indication that Bale will remain at the club beyond the September 2 transfer deadline, per Tom Allnutt of the Agence France-Presse:
Following Cristiano Ronaldo's departure last year, Bale had the opportunity to lay down a marker in the Spanish capital.
The Welshman returned 14 goals and seven assists in 42 appearances in what was a difficult season for the team, who finished third in La Liga and were knocked out of the UEFA Champions League in the last 16.
It was the first season since the 2014-15 campaign that he managed to play more than 40 games, as he's been beset by injury issues in recent years.
As sports writer Andy West noted, his fitness struggles have affected his value:
There is little doubt that he is a world-class player when on form, though.
Alone he may not be able to fill Ronaldo's considerable shoes, but if he can get back to that level and play a similar or greater number of games than last season, he can still be an enormous asset to Los Blancos.