European governments are under mounting pressure to reopen their airspace after carriers said test flights had shown there was little risk from volcanic ash hanging over the continent.
British Airways said on Monday that the closure should be lifted after it detected no problems during a test flight through the debris from Iceland's Eyjafjallajokull volcano, which erupted last week.
"The analysis we have done so far, alongside that from other airlines' trial flights, provides fresh evidence that the current blanket restrictions on airspace are unnecessary," Willie Walsh, the BA chief executive, said on Monday.
The head of the International Air Transport Association (Iata) also criticised the flight ban, which has seen tens of thousands of flights cancelled and millions of passengers stranded around the world.
"We are far enough into this crisis to express our dissatisfaction at how governments have managed the crisis," Giovanni Bisignani told reporters in Paris.
"It took five days to organise a conference call with the ministers of transport," adding later in an interview with BBC radio: "This is a European embarrassment and it's a European mess."
Naval ships dispatched
But Siim Kallas, the EU transport commissioner, said there could be "no compromise on passenger safety", adding that the responsibility for travellers must lie with the airlines.
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"Airlines have to provide necessary information to care and to provide care and to reimburse or re-route if necessary," he told a news conference in Belgium.
Gordon Brown, Britain's prime minister, who said he would be looking at the results of the BA test flights, has reacted to the crisis by dispatching several Royal Navy ships to rescue stranded Britons abroad.
Brown said two ships, the HMS Ark Royal and HMS Ocean, would be setting sail on Monday, while assault ship HMS Albion was already en route to Spain, to pick up British troops returning from Afghanistan.
The proposal has caused a stir in the UK where a general election campaign is under way, with the opposition leader David Cameron seeking credit for the idea.
"The idea of using the Royal Navy was actually something the Conservative Party very constructively suggested and I am delighted the government has taken it up," he said.
London's Heathrow airport, one of the first to close due to the volcanic ash cloud, is gearing up to reopen at 21:00 on Tuesday, a source told Al Jazeera.
Imran Khan, Al Jazeera's correspondent at the airport, said it would be a "glimmer of hope" for passengers.
He added that Scotland is opening its airports on Tuesday morning, a sign the UK's airports may be gradually resuming operations.
Fighter jets 'damaged'
Nato has also been involved in the debate over potential risks posed by the volcanic ash, with reports that two fighter jets suffered engine damage after flying through European airspace.
A senior Western diplomat said glasslike deposits were found inside the plane's engines.
"This is a very, very serious matter that in the not too distant future will start having real impact on military capabilities," the unnamed US official said.
But Anders Fogh Rasmussen, Nato's secretary-general, said the volcano had no effect on the military alliance's operations and had no information on damaged fighter jets.
Meanwhile scientists have said Iceland's volcano is spewing out less ash, a sign that Europe's air traffic crisis could soon be over.
"Currently the eruption has diminished markedly," Bryndis Brandsdottir of the University of Iceland told the AFP news agency, based on seismological radar readings.
She said the eruption started decreasing Sunday afternoon, but it had been difficult to monitor the volcano at the time "because of weather and clouds."
David Chater, Al Jazeera's correspondent in Reykjavik, Iceland's capital, said it appeared there was a fresh lava flow coming out of the volcano, which experts believe is "good news".
"What it means essentially is that the ash itself has formed a cap around the eruptions, which means the ice from the glacier is no longer coming into explosive contact with the lava, with the magma.
"So that phase [of the ash eruptions] they think is now over. But they don't know what is going on below the surface," he said.
But weather experts say wind patterns meant the ash plume is not likely to move far until later in the week, and that a shift in jet stream winds from Thursday could flush it out of most of Europe.
A number of airlines have carried out test flights since the eruption, with Germany's Lufthansa making 11, Dutch carrier nine and Air France seven.
KLM said most European airspace was safe despite the plume of ash, reporting no damage to engines or evidence of dangerous ash concentrations after the test flights.