By Lionel Laurent and George Georgiopoulos

LONDON/ATHENS (Reuters) - Greece's cash crunch has pushed several European brokers to cut back on Greek share trading or restrict approval of new positions in case Athens imposes capital controls, industry sources told Reuters.

The measures vary from broker to broker, rather than reflecting industry-wide moves. Trading sources in Athens said global banks were still routing Greek trades, while the Athens stock market operator said the proportion of turnover involving foreigners was little changed last month from a year earlier.

Nevertheless, the caution of some brokers indicates rising awareness about the risk of having unsettled trades or cash with a Greek counterparty, as talks grind on between Athens and international creditors about the release of around 7.2 billion euros ($8 billion) in aid.

Greek Finance Minister Yanis Varoufakis said last week that capital controls had not been considered, even though his government is struggling to meet its international debt repayments and domestic expenses without the bailout money from the European Union and IMF.

These problems have raised fears that Athens might try to prevent funds from leaving the country in the event of a major flight of capital, as Cyprus did during a crisis in 2013.

Most equities trades go through central clearing and settlement, which offers guarantees. But brokers appear worried that cash balances they keep at Greek banks to facilitate trades could get stranded in the accounts, or it could be difficult or impossible to complete deals promptly.

"The risk of capital controls in Greece has increased over the past few days ... We are now making sure that any new trading lines go through several levels of approval and compliance," said an executive at a top-5 EMEA equity brokerage who declined to be named.

A spokesman for Dutch online broker Degiro said the company told clients last week it had stopped offering finance for trading in Greek securities and for borrowing stock for short-selling. "The outcome of the current political and economic situation in Greece is unforeseen ... These measures are deemed necessary in order to reduce risk and exposure," the broker said.

A spokesman for Paris-based Kepler Cheuvreux, in which French bank Credit Agricole <CAGR.PA> owns a 15 percent stake, said the broker had also taken measures related to trading Greek securities "over the past few weeks" but declined to comment further.

This is still some way from the financial industry's widespread unwinding of Greek exposures that began at the height of the euro crisis in 2011 and 2012. Several sources at other brokerages in London and Frankfurt said there had been no change to the way they traded Greek securities, and investors unable to use the cautious brokers may simply be turning to others.

Citigroup <C.N>, UBS <UBSG.VX>, BofA-Merrill <BAC.N>, Deutsche Bank <DBKGn.DE> and Societe Generale <SOGN.PA> - among the top foreign brokers on the Athens exchange - all declined to comment.

The head of Athens stock market operator Hellenic Exchanges, Socrates Lazaridis, told Reuters that foreign investors had accounted for 64.8 percent of turnover last month versus 64.9 percent in April 2014.

"The data do not show that the trading activity of foreign investors on the Athens stock exchange has fallen," he said.

But even those brokers that have not taken extra measures said they were constantly monitoring their exposure to Greece.

"We are cautious and we are monitoring the situation, even if we are still trading in Greece," said a London-based source at another top-5 EMEA equity brokerage.

(Reporting by Lionel Laurent and George Georgiopoulos; Additional reporting by Anthony Deutsch in Amsterdam, Alexandre Boksenbaum-Granier in Paris, Francesco Canepa in Frankfurt, Vikram Subhedar in London and Sudip Kar-Gupta in London; editing by David Stamp)