BRUSSELS (AP) — The European Union is ready to offer Turkey new incentives to better tackle the Syria refugee crisis, including money, the easing of visa restrictions and better intelligence sharing.

The offer came in an action plan unveiled Tuesday, which in exchange, would see Turkey improve its asylum and documentation procedures and beef up border security.

Around 2 million refugees from Syria are currently in Turkey, and tens of thousands of others have entered the EU via Greece this year, overwhelming coast guards and reception facilities.

The "Draft Action Plan" was presented to Turkish President Recep Tayyip Erdogan during his two-day visit to Belgium and the EU, but has yet to be officially accepted by Turkey.

It doesn't address demands made by Erdogan earlier Tuesday for Turkey's EU membership process to move ahead more quickly. Nor does it directly address calls he made Monday for European backing for the creation of a safe haven and no-fly zone around Syria's northern border, which he said are key to ending the refugee crisis.

However, there could be some wiggle room in the EU pledge to provide assistance aimed at Syria refugees to ensure "the weakening of push factors forcing them to move towards Turkey."

Under the offer, Turkey would receive up to 1 billion euros ($1.1 billion) to help manage its refugee crisis, and EU funding to help build six reception centers for refugees in Turkey.

By accepting the plan and implementing it, Ankara "would also contribute to accelerate Turkey's fulfilment of the visa liberalization roadmap benchmarks," the text said, referring to Ankara's long-held desire to smooth travel for its citizens.

The EU is caught in a delicate balancing act, wanting to encourage Turkey to better control its borders amid continued criticism of the abuses of the Kurdish minority there and attacks on the media and justice system.

"If we want to cope with this problem, Turkey is absolutely a key partner," European Council President Donald Tusk told EU lawmakers Tuesday.

"I know that this is a very dramatic dilemma," he said. "We have to try to cooperate with Turkey because in fact we have no other options."

During his two-day visit to Brussels, Erdogan warned the Europeans that many more people are likely to flee northern Syria.

Tusk said that "according to Turkish estimates, another 3 million potential refugees may come from Aleppo and its neighborhood."

He warned that "the world around us does not intend to help Europe" and that some of the EU's neighbors "look with satisfaction at our troubles."

Tusk also lashed out at countries for failing to fully respect EU asylum and border rules, naming Hungary, Italy, Slovakia and Greece.

"We have to respect commonly agreed rules," he said, adding that when countries say they intend to flout the laws "they undermine the essence of solidarity and our community."

Meanwhile, Austrian Chancellor Werner Faymann arrived on the eastern Aegean island of Lesbos with Greece's prime minister to see the impact of the refugee crisis and to examine facilities set up to handle the thousands of people who arrive daily.

Faymann and Greek Prime Minister Alexis Tsipras went to the island's main port Tuesday, where a ferry was preparing to take about 2,500 people to Athens. They were later to tour a reception center set up to register and process arriving refugees and migrants.

About 400,000 people have reached Greece so far this year, most in small overcrowded boats from the nearby Turkish coast. Most arrive on Lesbos.

In Britain, the interior minister said she wouldn't agree — "not in a thousand years" — to a common European immigration policy to deal with the flow of migrants and refugees coming to the continent.

Home Secretary Theresa May told the Conservative Party conference Tuesday that other European countries should also toughen up, arguing that in the last few years more people had applied for asylum in the EU from Balkan countries — which have not seen war for years — than from Syria.

She said the crisis "can only be resolved by nation states taking responsibility themselves — and protecting their own national borders."

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Elena Becatoros in Athens, and Jill Lawless in London, contributed to this report.