The regulator alerted organisations to the scam today and urged those affected to get in touch.
"A number of charities have been invoiced fraudulently, although we do not believe that any have paid money," a spokesman for the regulator said.
He said the spoof messages came from an email address not registered to the regulator’s @fundraisingregulator.org.uk domain and contained a link to an external website.
The regulator had alerted the national fraud centre Action Fraud to the phishing attempt, the spokesman said, and was raising it with sector bodies including the Charity Commission, the Institute of Fundraising and the National Council for Voluntary Organisations.
The regulator has asked charities that spend more than £100,000 a year on fundraising to pay a voluntary levy to cover the Fundraising Regulator’s costs, which includes the operation of the FPS.
The regulator started the FPS this year to allow people to block post, telephone, email or text communications from named charities.
The amount that charities have been asked to pay by the regulator for the levy varies according to how much each organisation spends on generating voluntary income.
The spokesman added: "The Fundraising Regulator will never issue an invoice for the FPS, since it is funded by our levy. If you know or you believe you have received one of these fraudulent FPS emails and have concerns, or if you have any questions, please get in contact with us on 0300 999 3407 or at email@example.com.
"We would also like to ensure that those organisations that fall within the scope of the fundraising levy do not assume the invoices we are currently sending for year two are in any way associated with this fraudulent scheme."