The former British Open champion suffered the injury while on holiday in the Bahamas last week, but it was expected that he would be fit to return at Augusta.
However, Clarke, who missed the cut last year, will reportedly continue his treatment in the Bahamas.
"It is with deep regret that I will not be able to play at Augusta this year," Clarke said in a statement.
"Playing in The Masters is one of golf's greatest pleasures and I am very disappointed to be missing out."
The strain kept him out of the Texas Open last week.