"This work will help determine whether changes to the design, use or calibration of the regulatory capital framework may also be needed to ensure resilience against these risks," the BoE said in a statement.
It said it would give an update in 2022 after more research and a climate change and capital requirement conference.
Climate risks to insurers come from a reduction in the value of assets such as property due to weather events like floods.
Banks, meanwhile, could be affected by sudden falls in the value of assets like stocks and bonds they hold as the economy transitions to net-zero emissions.
Climate activists want regulators to go a step further and use capital requirements to penalise banks that finance fossil fuels, but the BoE says its remit is limited to ensuring banks and insurers stay stable in the face of climate change.
The BoE said in 2019 that banks and insurers should set out by the end of 2021 how they manage risks from climate change and disclose them.
The central bank said on Thursday that the firms it regulates have made "tangible progress" in meeting these expectations, but some are materially more advanced than others.
It also signalled a shift in gears next year in how it supervises and enforces these requirements.
"As we move into 2022, the PRA will actively supervise to ensure firms meet expectations, with firms needing to demonstrate a good understanding and management of climate-related financial risks on an ongoing basis," it said.
"We will consider the use of our full supervisory and regulatory toolkit to provide the necessary assurance or remediation where appropriate."
Enforcement can range from warnings to mandatory capital top-ups and even fines.
Separately, the Financial Conduct Authority in its own report said it will scrutinise how well the environmental, social and governance (ESG) characteristics of products align with the claims firms make on sustainability.
"Increased capital requirements for some banks and insurers seems highly likely and enforcement action by FCA is a clear threat," said Paul Edmondson, a financial services partner with law firm CMS.
The Basel Committee, which writes global rules on capital requirements applied by top financial centres, is also due to publish an update on whether climate buffers are needed.
The BoE, a member of Basel, is expected to be aligned with the global work.
(Reporting by Huw Jones; Editing by Mark Potter, Jon Boyle and Alexander Smith) ((email@example.com; +44 207 542 3326; Reuters Messaging: firstname.lastname@example.org))