Speaking to BBC Radio 4, Cunliffe said:
“It is a very volatile situation, it is a very fluid situation and I think it is a very dangerous situation in Greece now.
“The banks are shut, the economy is pretty much frozen. This is fast-moving, volatile and fluid. We have to prepare for the worst.
“We are not seeing signs of real pressure on the peripheral economies — Portugal, Spain, Ireland — in the way that we saw two years ago.
"But if there is pressure there, then the European Central Bank now has instruments it didn’t have in 2011 and 2012. They have said they will use those instruments. We in the UK have made our systems more robust. The defences are there, but if this happens, those defences will be tested.”
Greece failed to repay a €1.5 billion loan to the International Monetary Fund this week, and faces a referendum on Sunday to decide whether to continue with repayment plans. Several European leaders have warned that a "no" vote could see Greece leave the eurozone.
