Chris Coleman's side go into their final two matches sitting in second place in the table, a point ahead of Ireland who they face on the final day in Cardiff.
The 28-year-old missed their La Liga game with Espanyol on Sunday but reported to the Wales camp the following morning.
Bale sustained a calf muscle strain against Borussia Dortmund.
Gareth Bale out of Ireland game
After so many games at the start of the season and four months of injury, he was suffering after the game in Dortmund ". Despite Bale's absence, Bradshaw could well be kept out of the starting line-up by Liverpool teenager Ben Woodburn .
Tom Lawrence insists Wales will not be fazed by the loss of talisman Gareth Bale for their final two 2018 World Cup qualifiers.
It is now 18 different injuries for Bale since his arrival from Tottenham in summer 2013 - the majority of his absences have been due to his problematic calf muscles (six to the left, two to the right) - but there have also been back, knee, ankle and hamstring issues.
"When they lose a player like that, they'll just want to win even more and show people that they don't just need him in the team to win games".
His absence could mean a first global start for Woodburn, whose goal off the bench against Austria in September catapulted him onto the big stage.
The 17-year-old striker scored on his debut as a second-half substitute against Austria and then set up the opening goal in a 2-0 win in Moldova three days later. If Woodburn can help lift Wales into the World Cup playoffs, it will only be more fuel for the runaway hype train.