The Bank's deputy governor for financial stability said it was "focused very, very closely on this".
"We're monitoring the plans of the financial institutions in the UK; their plans for how they are going to deal with Brexit, and we're also monitoring the plans of all of the European Union firms that operate in the UK and the City of London to look at their plans for how they're going to manage Brexit," Cunliffe told Sky News.
"And if we start to see financial stability risk coming out of those plans, the way they think Brexit will affect them, then we'll start to take action to monitor that."
Much has been made of potential moves in the City, with many banks mulling staff moves from Britain before it leaves the EU in March 2019.
Cunliffe also discussed the financial crisis a decade on from Northern Rock's crash, noting banks will always have to take risks and could still fail, but stressed that that the system was "safer, simpler and stronger than it was before the crisis".
"Banks have to take risks, and those risks will sometimes go wrong; banks will make losses, but you have to make sure that when they take those losses they can absorb them without just falling over," Cunliffe said.
He added that bank capital in the system was around 10 times what it was before the crisis, and it was not just about quantity, but quality. "It has to be real money, basically shareholders' funds that can absorb losses when loans and other things go wrong," Cunliffe said.
He also said that the Bank of England is setting up a regime so if a bank fails "it can be wound down safely and the losses can be borne by the people who profit from the banking good times, and not by the taxpayer".