"The role that is always required from the European Union, past, present and future, is financial support for the national authority," Abbas said.
"We want the European Union to have a political role that compliments the role of the quartet in the region."
Abbas said on Tuesday he is willing to hold a dialogue with any prime minister the Israelis choose as long as the leader does not "bring us back to the starting point".
"There will not be other opportunities to make peace in the Middle East if we let this moment pass," he said.
"This is why we are telling anyone who could be the next Israeli prime minister, in the same way as we are telling President [Barack] Obama and President [Nicolas] Sarkozy, that this is an opportunity that we cannot miss."
Abbas was speaking at the start of a European tour which will include visits to Britain, Italy, Turkey and Poland.
Kouchner said Abbas had discussed working with Hamas, Fatah's rival Palestinian faction, during their talks.
The French foreign minister said: "The president and I talked about the Palestinian reconciliation and that is a major issue, that would leave open the possibility to find an agreement to work out a Palestinian unity government to be put in place.
"First of all, in Gaza, the gates must be opened. The checkpoints must be opened. This is not sufficiently the case right now."
Nour Odeh, Al Jazeera's Palestinian West Bank correspondent, said that Abbas "wants European assurances that they will deal with a Palestinian government of national unity that will include Hamas, no doubt, if and when that government is formed".
"In 2007, the Europeans did not agree to deal with a government of national unity headed by Hamas. So Abbas does not want a repeat of that," she said.
"He is telling the Europeans that it is simply unacceptable for them to adopt that same position because it literally jeopardises the future of Palestinians and their aspirations to unite."
Reconciliation between Fatah and Hamas would be predicated on "two essential principles", Abbas said.
They are "a national unity government whose formation will not lead to a blockade [and] a national unity government that considers itself bound by international legality and previous agreements," he said.
"If Palestinian reconciliation is achieved, if the US administration is ready for work and the Israeli government is formed in a short period, I believe that the conditions would make it possible for the political process."
While in Paris, Abbas also met Sarkozy, the French president, and Bernard Accoyer, the president of the lower house of France's parliament.