The BoE and ECB declined to comment.
The move also risks further inflaming tensions between the UK and Europe amid delicate political negotiations over their post-Brexit relationship for financial services.
Last week, Britain and the EU agreed to set up a new "talking shop" on regulatory co-operation, but the EU has still not indicated whether it will grant "equivalence" status to British regulations. Under this mechanism, the EU would recognise UK financial rules as equivalent to its own and vice versa, which restore to London a degree of the direct access to the bloc it lost after the split.
Banks that are deemed to be taking unnecessary risk can have their regulatory capital requirements increased by the BoE's Prudential Regulation Authority. The new BoE requirements not only apply to UK and EU banks but also those from third countries such as the US or Switzerland. In the past they used London as a base to "passport" their financial services into the continent.
Brexit meant that the UK lost this unrestricted access and as a result many lenders set up or bolstered their operations inside the EU, notably in Paris, Frankfurt and Amsterdam.
While London has lost big chunks of euro-denominated share and derivative trading to Amsterdam, there has not been the exodus of jobs or revenue from London that some had predicted. EY estimates only 7,600 UK jobs and £1.3 trillion ($2.5t) in assets have been transferred to the continent.
EU regulators have insisted that banks cannot operate small "brass-plate" entities inside the bloc, with top executives, risk-taking staff such as traders and compliance officials remaining in London.
The BoE's Prudential Regulation Authority accepted this stance and in the run-up to Brexit required lenders to draw up detailed models for what their businesses would look like initially and over a longer period of time. These were known as "day 1" and "day 2" plans.
However, the BoE now believes that the EU is trying to force banks to go beyond the agreed "day 2" planning, the people familiar said. The PRA is concerned that "ad hoc" requests to transfer more business and leadership could undermine the safety and soundness of lenders' London-based operations.
One example being examined is that the SSM has suggested to some large American banks that they should only operate one trading desk to handle all transactions of a particular financial product within the same timezone, one of the people said.
This may stem from the ECB's push to stop banks making excessive use of a controversial technique known as "back-to-back" operations that lets them transfer the risk of EU deals to a parallel transaction in the UK.
One person briefed on the matter said the ECB was not seeking that banks do more than stick to their Brexit plans under the "target operating models" agreed with supervisors.
Andrea Enria, chair of supervision at the ECB, told the European Parliament last week that supervisors were still in discussions with banks "to ensure that in the post-Brexit world they allocate enough staff and assets to institutions inside the banking union." He said this was "necessary to ensure adequate management of risks both in and from Europe".
- Additional reporting by Martin Arnold.
Written by: Stephen Morris and Caroline Binham
© Financial Times