Assuming these are agencies contacting you, there are reasons why they would need your SSN. however, background checks require you written permission and happen only after there is a job offer. Do not give out personal information on an initial inquiry. That is a red flag for identity theft.
We have several clients who use a "job portal" that only agencies such as ours have access to. When a vacancy is released on the job portal, there was at one time a problem with immoral agencies submitting resumes wholesale without ever speaking to a candidate. Some clients dealt with this by cutting out these agencies, others by implementing "ensured contact" methods - requesting references, a written statement of representation by the candidate, or detailed personal information such as birth date and SSN.
I can't think of any reason a recruiter would need this information unless there is a specific role at one company that the recruiter has spoken to you about. If a recruiter calls you and asks outright, in the first call, without any mention of a job vacancy or client, then no, I would not give your SSN. At the very least, I would ask why they need it at such an early stage. However, if your recruiter has told you about a vacancy that you're interested in and has told you that the SSN is a prerequisite for submission, you will not be submitted unless you provide it. Other candidates will, and other candidates will get the job.
Obviously, do your due diligence and ensure a recruiting company is who they say they are, and you're not just speaking to a particularly clever scam artist.