Terror suspects have been arrested across the EU
The European Union has made public a list of 47 requests by the United States for co-operation in combating terrorism.
EU officials stressed their desire to work closely with the Americans but they said the use of the death penalty in the US stands in the way.
The requests came in a letter to the European Commission President, Romano Prodi, and Guy Verhofstadt, the Prime Minister of Belgium, which now has the EU presidency.
Most pose no problem and some - such as freezing the assets of wanted terrorist suspects - have already been agreed.
But EU officials say some of the requests may be impossible to meet.
In particular, the US use of the death penalty could be a reason to refuse to extradite terrorist suspects.
EC spokesman Jonathan Faull says that EU states are signatories of the European Convention on Human Rights which outlaws the death penalty.
"I think our starting positions - some of which are very firmly held positions of principle enshrined in our laws and reflecting our values - are well-known. There are complicated issues. They have to be worked through and we are seeking to do that," he said.
The Americans have asked for an end to discrimination against US extradition requests or for suspects caught in Europe to be quickly deported to America.
And it is not just theoretical.
In Britain, three men accused of involvement in the bombing of US embassies in Africa three years ago are now fighting extradition to the US.
And Germany says some members of the al-Qaeda network that is thought to have carried out the attacks on New York and Washington, may be at large in Germany or elsewhere in Europe.