The agreement last year among 140 financial centres was aimed at raising standards of banking supervision and is closely linked to an international campaign against money-laundering.
Michael Foot, the Bank's director responsible for supervision, told a crime prevention conference in the Cayman Islands that reputable centres might refuse to allow dealings with countries that did not abide by the spirit of the agreement.
Banking supervisors from the main industrial countries last year reached a co-operation agreement with the Caymans and 18 other offshore centres including Guernsey, Jersey, Gibraltrar and the Bahamas, aimed at improving the flow of information.
This was endorsed at a meeting in Sweden by representatives of140 financial centres, which agreed to bring their own procedures up to the same standards as the original group.
Mr Foot warned that countries which "do not commit wholeheartedly to putting into effect the spirit of the accord which has been worked out so painstakingly may quickly find themselves regarded by others as beyond the pale".
He said that since last July, a European Union bank can lose its licence if it has close links - as parent, subsidiary or sister company - with a financial centre whose secrecy laws prevent effective banking supervision.
Mr Foot called for similar cooperation in the fight against money-laundering, and complimented the Cayman Islands on setting new standards with the enactment of legislation to counter its use to hide the rewards of crime.