Sandwell Council said this includes "several individuals aggressively begging" in Bearwood, located in Smethwick.
The local authority responded after Abbey ward councillor, Dr Ann Jaron, revealed the issue has been "long-standing" in Bearwood.
A council spokesman said more than 70 genuine rough sleepers have also been helped into emergency accommodation since the Covid-19 pandemic began.
The spokesman said: "The council does significant work to assist people who are homeless or sleeping rough.
"Sandwell Council’s housing solutions service has helped more than 70 rough sleepers into emergency accommodation since the start of the Covid-19 pandemic and a number of these now have a home to call their own.
"However, in Bearwood we have also taken action against a number of individuals who are not rough sleepers but who are in fact falsely presenting themselves as being homeless or destitute.
"This has included several individuals aggressively begging in Bearwood, one of which who is subject to an anti-social behaviour injunction and two who have been issued with pre-court warnings.
"Our ASB [anti social behaviour] team continues to work with our partners including West Midlands Police, the environmental protection team and housing solutions to tackle ASB in our town centres."
The local authority has asked anyone who sees incidents of aggressive begging taking place to report the issues.
The spokesman added: "Anyone who witnesses aggressive begging in Bearwood or any anti-social behaviour anywhere in Sandwell should report it at www.sandwell.gov.uk/asb.
"Anyone who is homeless, or anyone who spots someone sleeping rough, should get in touch with the council at www.sandwell.gov.uk/homelessness or by calling 0121 368 1166, option two."