There is no way for individual citizens to check the citizenship status of a particular Social Security number, but there are free ways to check the owner's work authorization. U.S. Citizenship and Immigration Services offers a free tool to check an individual's employment eligibility.
The online tool provided by the government is intended for use by the holder of the Social Security number, not by third parties. Employers must use the employment verification tool provided by the USCIS instead. These verification tools tell whether an individual has temporary or permanent work authorization in the United States. It is impossible to discern between a citizen and a permanent resident with this method because both citizenship and a green card confer permanent work authorization.
Checking an individual's physical Social Security card reveals the same information. Cards issued to citizens and permanent residents are identical, but cards for temporary residents reflect the holder's work authorization. Cards for residents with temporary work visas are labeled as being valid with DHS authorization only. Cards for residents without work authorization are clearly labeled as being invalid for employment purposes.
There is no need for individuals or corporations in the private sector to determine whether a Social Security number belongs to a citizen or a permanent resident. The only restrictions placed on permanent residents are government ones, such as a lack of voting rights and the inability to hold certain government jobs.