Bale joined the Spanish giants at the start of the month for a reported world-record fee of £85m.
"It's probably a relief for him, I think," Sigurdsson told Fotbolti.net.
"The players aren't thinking too much about this but it's great for him to join one of the world's biggest clubs.
"He's a great player with a lot of pace and sharpness, and I just hope he starts doing well as soon as possible and fits right in."
Sigurdsson is currently with the Iceland squad as they prepare for Tuesday's World Cup qualifier against Albania.