London Capital and Finance (LC&F) went into administration in January 2019 just a month after concerns were raised about the provider by the FCA.
The FCA directed LC&F in December 2018 to withdraw its promotional material for its mini-bonds on the basis that the marketing was "misleading, not fair and unclear."
While London Capital and Finance and the promotional material were regulated by the FCA, the mini-bonds were not.
Today (March 19), Ms Morgan said the FCA needed to look into whether mini-bonds should be under "the perimeter of regulation."
HM Treasury has the power to require the regulator to conduct such investigation into whether there was regulatory failure in handling London Capital and Finance.
Ms Morgan has also written to John Glen, economic secretary, to urge HM Treasury to use this power if the FCA declines to investigate.
Her letters to the FCA's board and Mr Glen comes just a day after it was revealed four individuals associated with collapsed mini-bond provider London Capital & Finance PLC are being investigated by the Serious Fraud Office.
In a statement the SFO confirmed the four individuals were arrested in the Kent and Sussex areas earlier this month and were released pending further investigation.
The investigation was opened following a referral from the Financial Conduct Authority to the National Economic Crime Centre, and the SFO said it had been working in conjunction with the regulator since.
Ms Morgan said: "The FCA is currently investigating LC&F's marketing material and the SFO is investigating individuals associated with the company.
"Yet there is a broader need to understand what can be learned in a regulatory sense from the events at LC&F.
"I have therefore requested that the FCA board consider whether the failure of LC&F, the potential harm to those consumers involved, and the regulatory system that led us here, warrants a statutory investigation.
"If the FCA decline, I have asked HM Treasury to consider using its power to require the regulator to conduct such an investigation.
"Even if the regulator does not conduct an investigation, the FCA board should set out whether firms are using their FCA-authorisation in a way that may be misleading to consumers, whether consumers need greater clarity on what such an authorisation does to protect them, and whether mini-bonds should now be regulated.
"The stories of those affected by the actions of LC&F are distressing. The government and the regulator must do all they can to prevent history from repeating itself."
A spokesman for the FCA said: "We have received the letter and will respond."