Woods responded to newly-appointed chair of the Treasury Committee Nicky Morgan's request on 2 August, following an extensive survey of financial services firms, which received 401 responses operating across borders on their preparedness for Brexit.
According to Woods, this would make such firms more complex and hard to supervise because of strong UK-EU interconnections.
The warning came in a letter from head of the PRA and deputy governor of the Bank of England, Sam Woods, following the PRA's recent survey of financial firms that undertake cross-border activities between the United Kingdom and EU.
The PRA faces having to authorise and supervise a significant number of additional firms, which could place a material extra burden on resources, Woods said.
CEO Sam Woods has warned about the dangers of a cliff-edge Brexit and the need for a period of "transitional arrangements" if no deal is reached.
The risks of financial instability primarily arise from the possibility of fragmentation of services, which would increase costs and risks, and the possibility of disruption to the United Kingdom economy.
Officials have also noted that any wider harm to the economy following Brexit could expose banks and other financiers to higher loan losses and lower asset prices.
Woods did not provide details of the individual plans the Bank had demanded by 14 July. For instance, RBS is preparing to expand in Amsterdam; Barclays and Bank of America are moving staff to Dublin; while Morgan Stanley has picked Frankfurt.
The Bank of England said a transition period after Britain leaves the European Union would give banks more time to make orderly changes as Brexit poses risks to financial stability.
Woods told Morgan the Bank would "continue to develop and refine its assessment of the potential stability risks associated with the withdrawal" from the EU.
However, the Close Brothers research which was carried out by YouGov, reveals that 11% of adults are likely to increase the frequency with which they receive financial advice since the United Kingdom voted to leave the European Union as they look to ensure that they are financially prepared for the possibility of bumpy years ahead.
The Bank may end up regulating more firms if there is no deal on passporting, the system through which some insurance companies and banks based in the EU, Iceland, Liechtenstein and Norway are able to operate in the United Kingdom without Bank of England authorisation. It has warned it may need to ask for more if the biggest risks around Brexit crystallise.