The top lending officer of a major U.S. bank joked recently that the institution's two officers who ran the Cayman Islands desk had a brochure pointing out the pleasures of the Caribbean paradise.

"They've never been there," he said. Now it is likely they never will, at least on the bank's money.

After this particular bank and scores of other U.S. banks open so-called international banking facilities (IBFs) Thursday, the role of the Caymans and other off-shore banking centers like the Bahamas probably will shrink considerably.

For the first time in history, U.S. banks--as well as the U.S. branches of foreign banks--will be able to play in the huge international money game from home. The new facilities will be free of many of the federal regulations that forced U.S. banks to conduct much of their overseas business in branches located abroad. And many states have obligingly passed new legislation that frees profits of these facilities from state and local taxes.

Even the District of Columbia, hardly a major international banking center today, is toying with exempting the earnings of international banking facilities from local taxes.

The new facilities are designed to compete with the so-called Eurodollar market, a pool of $1.56 trillion of currencies that are bought and sold, or loaned and deposited, in foreign countries. About two-thirds of the Eurocurrencies are so-called Eurodollars.

A Eurodollar is a U.S. dollar that is owned by a nonresident of the United States, whether a company, an individual or a foreign bank. Eurodollars are free of U.S. regulations, and they finance a good chunk of international transactions.

The facilities were approved last spring by the Federal Reserve Board, the nation's central bank, after years of lobbying by such major New York institutions as Citibank.

Federal regulators and bankers have estimated that within a few months IBFs could have as much as $80 billion to $120 billion in loans and deposits, with about half the business coming from domestic banking offices and half from foreign offices.

The new facilities will mean nothing to U.S. companies or individuals. The so-called IBFs will be able neither to raise deposits in the United States nor to make loans to U.S. residents. Customers and clients will be confined to foreign governments, foreign corpo rations, foreign individuals and the foreign subsidiaries of U.S. companies. Subsidiaries of foreign companies doing business in the United States will not be permitted to bank at an IBF.

The banks will raise their deposits--which must be left with the bank for at least two days--in the Eurodollar market.

According to the Federal Reserve, U.S. banks and the U.S. branches of foreign banks have filed applications to open more than 200 international banking facilities in cities such as New York, Chicago, Houston, Miami and Los Angeles. More than half--116 at the latest count--will be opened in New York.

Of the 116 to open either Thursday or within the next few weeks in New York, 78 will be opened by branches or agencies of foreign banks doing business in the United States, 31 will be opened by New York-based banks (some of those, such as European American Bank are foreign-owned), and 7 will be opened by Edge Act corporations, which are special foreign banking subsidiaries of U.S. banks.

Chase Manhattan Bank, for example, will open four international banking facilities Thursday. One will operate in New York. The others will be based in Los Angeles, Houston and Miami.

Most of the early IBF business will come from within banks themselves, as the institutions move loans and deposits that technically are on the books of offshore branches--such as shell branches in the Cayman Islands or Nassau--to the U.S IBF.

The attraction of operating out of an IBF, rather than out of the Caribbean, is lower cost. Presumably because the risk of placing funds in the U.S. is lower than in the Cayman Islands, many depositors would be willing to accept a lower rate from an IBF than from a Cayman branch.

That such preferences may exist is suggested by modest interest-rate differentials between some existing banking centers in the Eurodollar market, according to Morgan Guaranty Trust Co.

U.S. banks also will be able to exercise greater management control over lending done through IBFs rather than through foreign branches, according to Robert E.L. Walker, vice president of Continental Illinois National Bank, Chicago's biggest and the second-largest commercial lender in the United States.

Despite the heady growth many banks see from the international banking facilities, it is doubtful, at least at first, that there will be much impact on major international centers like London, the seat of the Eurodollar market.