The organisation that facilitates the bulk of the world's cross-border payments is facing growing U.S. pressure to do what it has never done before - cut a country off from its global messaging system.

Belgium-based SWIFT - the Society for Worldwide Interbank Financial Telecommunication - is vital to international money flows, exchanging an average 18 million payment messages per day between banks and other financial institutions in 210 countries.

The member-owned cooperative has been described as the 'glue' of the global banking system with the value of daily payments using SWIFT estimated at more than $6 trillion (3.78 trillion pound).

You would struggle to find any bank or financial institution not connected to SWIFT. Non-financial users include General Electric, Google, Microsoft, Danone, Daimler and Sony.

The United States is seeking tougher scrutiny of banking transactions and oil shipment financing with Iran, which it accuses of seeking to develop nuclear weapons.

A U.S. Senate Bill, if it becomes law, would direct the White House to press SWIFT to drop Iranian banks, and would give the Treasury Department the power to sanction SWIFT and the banks that own it.

Nineteen banks and 25 connected institutions from Iran sent and received some 2 million messages in 2010. They included banks the U.S. accuses of financing Iran's nuclear programme or terrorism - Mellat, Post, Saderat and Sepah.

Faced with outside pressure, SWIFT's typical response has been: don't shoot the messenger. The Belgium-based body, involved in 80 to 90 percent of all global payments, is keen to point out that it does not carry out transactions and is only a messaging system, more akin to a telephone service, which U.S. lawmakers are not targeting. SWIFT does not hold accounts for members and does not perform clearing or settlement.

We are not a bank and do not hold funds. Our member banks are responsible for the content of these messages and complying with applicable financial sanctions; they are not monitored or controlled by SWIFT, the cooperative said in an email exchange.

However, it said last week that it was working with U.S. and EU authorities to resolve the issue.

This is a complex situation, it said, adding the impact on the global financial payments system and the flow of humanitarian aid to Iran required careful thought.

SWIFT has faced tests of its independence before.

After the United States and European Union had imposed sanctions on Myanmar in the 1990s over its human rights record, SWIFT faced NGO pressure to shut out the country's banks.

SWIFT's response then as it is now was that it is just a common carrier.

At SWIFT I didn't make the law. We just followed the law, said Leonard Schrank, who was chief executive from 1992 to 2007.

Over the years, SWIFT has faced various subpoenas to release information, but typically resisted, convincing investigators to target the responsible banks instead.

That changed after the September 11 attacks in 2001.

Within days I got a call from the Treasury, said Schrank.

Within weeks, SWIFT set up an automated programme allowing U.S. investigators to track the payments of suspected terrorists. Few people knew of the Terrorist Finance Tracking Program as it was subsequently called, until 2006 when The New York Times revealed its existence. It led to a wave of criticism from Europe about invasion of personal privacy.

Europe is starting to pay attention this time too. No formal talks have begun, but EU governments have begun informal discussions on whether to include SWIFT in its own package of sanctions against Iran. Some are privately concerned that Washington could go as far as threatening SWIFT itself if it does not shut out Iran.

We are forced to look at it when Congress does because of the consequences, said one senior EU diplomat, speaking on condition of anonymity.

Some of those pushing for tougher measures against Iran point out that SWIFT's rules allow it to expel users if they harm or threaten to harm the organisation's reputation. It therefore has the power to act, they reason.

The counter argument is that if SWIFT gives ground over Iran, it may well face calls to shut out other countries. China, for example, might want Taiwan excluded.

SWIFT, a cooperative of more than 10,000 users and with a board including executives from Citi, UBS and Deutsche Bank, says it is aware of the gravity of the situation around Iran.

SWIFT is clearly concerned about setting a precedent here, but we're looking at a Mideast war or worse. Hopefully they will find a solution, some way of ensuring this is just a one-off, said Schrank. Otherwise, SWIFT could get endlessly caught up in financial disputes as well.

(Additional reporting by Justyna Pawlak; editing by Janet McBride)