The FSA are concerned that 'recovery firms' are unlawfully targeting people who bought shares through a company that has gone into liquidation.
Investors who bought shares through certain companies have been cold-called by people claiming to represent recovery firms, the FSA has said.
They offer to either buy back shares at an attractive price or put investors in touch with a buyer, but only after an up-front fee has been paid.
The FSA said the offer is not genuine, and as soon as the fee is paid the so-called recovery firm representatives disappear with the money and without purchasing the shares.
It added that they are often calling from outside of the UK, are not authorised by the FSA and are not permitted to approach UK consumers to promote financial services.
The FSA said it is looking into the complaints. It has urged consumers to use its register to ensure they only deal with authorised firms, which means they have access to a complaints procedure and compensation if thing go wrong.
Sarah Wallace said, "It remains to be seen whether the so-called 'recovery firms' are the next vehicle for possible unlawful conduct following on from the 'boiler room' scams. The FSA helpfully provides information regarding share scams on its website."