Britain left the European Union last month and is seeking access to the bloc's financial market once a "business-as-usual" transition period ends in December.
The London Stock Exchange's LCH unit clears the bulk of euro denominated swaps but access to clients in the EU after December has yet to be worked out.
The EU has said any access would be contingent on Britain applying rules that are "equivalent" to those in the bloc.
Britain on Tuesday called for long-term equivalence but the EU said it would not guarantee permanent access.
For LCH, EU regulators must also satisfy themselves they can supervise the inner workings of the company given it handles vast amounts of euro-denominated contracts.
This would be done under a memorandum of understanding (MoU) between the BoE and EU regulators. As a last resort, the EU can require EU investors use a clearing house in the bloc.
Cunliffe said cooperation could include EU regulators having a "line of sight, joint discussions of risks and mitigation" at UK-based firms, but such arrangements must be practical, Cunliffe said.
"Effective supervision of systemically important firms in business as usual cannot be achieved with multiple hands on the steering wheel," Cunliffe said in a speech in Berlin.
"Firms need clear and consistent messages. This is true, a fortiori, at times of stress."
UK authorities fear the MoU will not make clear that as home regulator the BoE is the point of contact for LCH, and not to go directly to the firm, and that the central bank has the last word in a crisis.
Cunliffe said arrangements for shared supervision need to be worked out carefully, subject to agreed procedures and recognise the primacy of a financial firm's home supervisor.
"It is perhaps human nature on either side to want maximum control and minimum responsibility. But a stable and effective regulatory and supervisory relationship cannot be built on those lines," Cunliffe said.
Meanwhile, the BoE is updating an MoU with U.S. regulators for supervising LCH and U.S. peers like ICE and CME.
It is expected to make clear that the home regulator has final say, helping to put pressure on the EU to follow suit in its MoU with Britain.
By Huw Jones