Bale, who missed Real Madrid’s weekend league game against Espanyol, had arrived with the rest of the Wales squad on Sunday and attended an awards dinner in Cardiff.
“But following dialogue with Real Madrid, Bale was sent for a scan and results confirmed that he wouldn’t be fit to feature,” the FAW said in a statement.
“Barnsley striker Tom Bradshaw will now join up with the rest of the squad later today.”
Bale is the most high-profile player in the Welsh side and the country’s hopes of qualifying for next year’s World Cup in Russia are still hanging in the balance.
Chris Coleman’s team are second in Group D, four points behind leaders Serbia and one clear of Ireland in third. They play in Georgia on Friday and host Ireland the following Monday in their final qualifier.
Wales need to top the group to secure an automatic place in Russia but will go through to the playoffs if they finish second.
Bale had been doubtful after Real manager Zinedine Zidane reported the player was suffering a small calf strain.
The 28-year-old has scored 26 goals in 68 appearances for his country and was instrumental in their run to the semi-finals of Euro 2016.