I believe that most of the "free" APIs make their money with ads embedded in your messages, or sent as a separate message. In addition, messages most likely come from a common shortcode. If you agree to these limitations, as @anirvan commented, Google is your friend - here is just one example . It is unlikely that you will receive widespread international support (if you need anything).
You can also try email on the SMS route, as many carriers will receive email to a unique address and forward it to the user. Again, Google can help find these mappings . Please note that messages may not be in the desired format, and if you do this with any volume, you may be angry with carriers.
There are also inexpensive APIs - certainly not free, but it might be worth considering depending on your use case. Here are a few:
If your main concern is cost, I think you will find that Nexmo has the most competitive prices.
Disclosure of Information . I am an evangelist developer for Nexmo.