(Reuters) -U.S. strikes on Iran's nuclear sites could heighten the threat to American operators in the region, an organisation that monitors flight risks warned on Sunday, as some airlines cancelled flights to and from Dubai and Doha.

Airlines continued to avoid large parts of the Middle East due to missile exchanges between Israel and Iran, the latest upheaval to travel in the region.

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.

But following an early barrage of Iranian missiles, Israel reopened its airspace for six hours on Sunday to bring back those stranded abroad since the conflict with Iran began on June 13.

Israel's Airports Authority said so-called rescue flights to the country would expand starting on Monday with 24 a day from various destinations, "marking a significant step toward the gradual restoration of routine international travel." It said each such flight departing Israel could carry up to 50 passengers.

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 US military interests in the Middle East - either directly or via proxies such as Hezbollah," Safe Airspace said.

Meanwhile, flight tracking website FlightRadar24, said airlines maintained flight diversions around the region.

Its website showed airlines were not flying over Iran, Iraq, Syria and Israel. They 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.

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.

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.

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. "We continue to advise a high degree of caution at this time," it said.

RESCUE AND EVACUATION FLIGHTS

The BA and Singapore Airlines 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's carriers, El Al Israel Airlines, Arkia, Israir and Air Haifa, said earlier on Sunday they had suspended rescue flights bringing people back to Israel until further notice.

El Al said it would also extend its cancellation of scheduled flights through Friday and Israir said it had halted the sale of tickets for all flights through July 7.

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.

El Al, along with Arkia, Air Haifa and Israir, said they would operate at least 10 flights on Sunday from 1100 GMT.

Tens of thousands of Israelis and others who had booked tickets to Israel are stuck abroad.

At the same time, 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.

Japan said it had evacuated 21 people, including 16 Japanese nationals, from Iran overland to Azerbaijan. It said it was the second such evacuation since Thursday and that it would conduct further evacuations if necessary.

New Zealand said it would send a Hercules military transport plane to the Middle East on standby on Sunday to evacuate New Zealanders.

The government was also in talks with commercial airlines on how they could help.