Sir John told the BBC the UK banking system was now stronger, while banks had cut their exposure to Greece.
He added that the possibility of Greece leaving the eurozone "could not be excluded".
"We have to prepare for the worst," he said.
On Tuesday, the Bank's Governor Mark Carney said the UK's direct exposure to Greece was minimal.
"This is not a new set of risks or a new threat to the UK - we have had issues around Greece since 2010," Sir John told Radio 5 live's Wake up to Money programme.
"People have taken action already to protect themselves against these risks - the banking system is stronger, banks have reduced their exposure to Greece and the eurozone economy and they are carrying much more protection in terms of capital.
"In the eurozone, we are not seeing the situation of 2011-12 - economic growth is stronger and the European Central Bank and governments have put in place instruments to protect the eurozone from contagion.
"Financial markets are not showing there is contagion or spreading of risks to the periphery... we are not seeing signs of pressure on Portugal, Spain and Ireland."
Greece is holding a referendum on Sunday, where Greeks will be asked to accept or reject proposals made by the country's creditors.
Eurozone finance ministers have ruled out any further talks on a fresh bailout for Greece until the results of the referendum are known.
Sir John Cunliffe, the deputy governor of the Bank of England, has said Greece is in a "very dangerous situation", but says actions have been taken in the UK to prevent contagion.