By Jamie Freed and Steven Scheer

(Reuters) -Israel briefly reopened its airspace on Sunday and is expanding flight operations on Monday, aiming to help tens of thousands of travellers stranded by widespread cancellations across the Middle East after the U.S. attacked Iran.

An organisation that monitors flight risks warned on Sunday that U.S. strikes on Iran's nuclear sites could heighten the threat to American operators in the region. 

With Russian and Ukrainian airspace also closed due to war, the Middle East has become a more important route for flights between Europe and Asia, but flight tracking website FlightRadar24 showed empty space over Iran, Iraq, Syria and Israel. Airlines have chosen routings such as north via the Caspian Sea or south via Egypt and Saudi Arabia, even if these mean higher fuel and crew costs and longer flight times.

Air France KLM said it has cancelled flights to and from Dubai and Riyadh on Sunday and Monday. Singapore Airlines said it cancelled flights from Singapore to Dubai following a security assessment and British Airways cancelled flights to and from Dubai and Doha. Those cancellations were only for Sunday, but Singapore said other flights between Singapore and Dubai may be affected as the situation remains "fluid".

BA, owned by IAG, said customers scheduled to travel between now and June 24 to Dubai and Doha can rebook up to and including July 6, free of charge. 

Israel meanwhile is starting to find ways for citizens stranded abroad to get home and for foreign tourists to leave, if slowly. Israeli airline El Al on Sunday said it had received applications to leave the country from about 25,000 people in about a day.

Israel's Airports Authority said so-called rescue flights to the country would expand starting on Monday with 24 a day from various destinations, although each flight would be limited to 50 passengers. El Al said it would start servicing eight international destinations on Monday after getting the massive wave of requests to leave the country. Following an early barrage of Iranian missiles, Israel also reopened its airspace for six hours on Sunday to bring back those stranded abroad since the conflict with Iran began on June 13. 

Safe Airspace, a website run by OPSGROUP, said the U.S. attacks on Iran may increase risks to U.S. operators in the region.

"While there have been no specific threats made against civil aviation, Iran has previously warned it would retaliate by attacking U.S. military interests in the Middle East - either directly or via proxies such as Hezbollah," Safe Airspace said.

Missile and drone barrages in a growing number of conflict zones represent a high risk to airline traffic.    

In the nine days since Israel attacked Iran, carriers have suspended flights to destinations in the affected countries, though there have been some evacuation flights from neighbouring nations and some bringing stranded Israelis home.

Airlines are also concerned about a potential spike in oil prices following the U.S. attacks, which will increase the cost of jet fuel. [O/R]

In the days before the U.S. strikes, American Airlines suspended flights to Qatar and United Airlines did the same with flights to Dubai.

Safe Airspace said airspace risks could now extend to Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and the United Arab Emirates.

Israeli authorities opened the country's main airport, Ben Gurion near Tel Aviv, for rescue flight landings on Sunday between 1100 and 1700 GMT. The small Haifa Airport serving Israel's north was also open from 1100 to 1700 GMT.

Tens of thousands of Israelis and others who had booked tickets to Israel are stuck abroad and nearly 40,000 tourists in Israel are looking to leave, some of whom are going via Jordan's borders to Amman and Aqaba and others via Egypt and by boat to Cyprus.

(Reporting by Jamie Freed in Sydney and Steven Scheer in Jerusalem; Additional reporting by Tim Kelly in Tokyo, Sam McKeith in Sydney, Alistair Smout in London, Jack Queen in New York and Peter Henderson in San Francisco; Editing by Clelia Oziel, Giles Elgood and Lisa Shumaker)