In Britain, restrictions for Scotland, Northern Ireland and the north of England were to be lifted from 0600 GMT, before Heathrow airport was expected to open later in the day.
Send us your stories, pictures or video here.
"The volcanic eruption has reduced and the volcano is not currently emitting ash to altitudes that will affect the UK. Assuming there are no further significant ash emissions, we are now looking at a continuously improving situation," Nats said in a statement.
The latest weather office advice was that the "contaminated area will continue to move south," offering the possibility that restrictions to airspace above England, including London, could be lifted later on Tuesday.
"It is now for airports and airlines to decide how best to utilise this opportunity. Passengers should contact their airlines to find out how this will affect their travel plans."
France was to begin reopening some of its airports from Monday while continuing to ask pilots to respect a no-fly zone in the areas defined by EU aviation authorities.
Francois Fillon, the French prime minister, announced "a partial reopening of the airports north of the line between Nantes and Nice, which will allow a progressive return to normal traffic".
"Air corridors will open between Paris and airports in the south, notably Bordeaux-Paris, Marseille-Paris and Nice-Paris from 8:00am on Tuesday [0600 GMT]," he said.
Also starting on Monday, it was announced that passenger flights will be allowed to depart from Schiphol Airport in Amsterdam.
"We are taking the lead on this," Camiel Eurlings, the Dutch transport minister said.
"But from tomorrow there will also be lots of other airports that will start allowing flights."
Air travellers, due to fly into reopened airspace, are being advised to check the status of their flight before travelling to the airport.
In Germany, Lufthansa said it had received special permission to fly 15,000 passengers home to Frankfurt, Munich and Duesseldorf from Asia, North and South America and Africa.
A spokeswoman for Lufthansa, Europe's biggest airline in terms of passenger numbers, stressed that the 50 long-haul flights were exceptions and "not a return to normal flight service".
The easing of the restrictions came amid criticism from carriers, who said test flights had shown there was little risk from the volcanic ash.
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.
Under the EU agreement, aviation authorities would carve airspace above the continent into three zones:
Zone One: Closest to the ash cloud where air traffic would be completely banned
Zone Two: Where there would be partial restrictions on flights
Zone Three: Free of ash, where flights could resume completely
"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."
Al Jazeera's Imran Khan, reporting from London Heathrow, said the EU agreement seemed to signal an easing of the worst peacetime air travel disruption ever.
"It's good news for many if it turns out to be true but there is cautious optimism being shared among stranded passengers here.
"Many believe that the airlines simply haven't given them enough information to be able to make any decision," he said.
Naval ships dispatched
Gordon Brown, Britain's prime minister reacted to the crisis by dispatching several Royal Navy ships to rescue stranded Britons abroad.
| Millions of passengers have been stranded
due to flight cancellations [Reuters]
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.
According to scientists, 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."
However, Canadian airlines said that nine flights were cancelled on Monday due to volcanic ash, or the threat of volcanic ash, in Canadian airspace.
A spokesman for Air Canada said the winds later shifted and normal operations resumed.