EU leaders are struggling to find a simple answer.
The official policy -- Prodi told RFE/RL this week -- is to avoid talk of doors altogether.
He has a clear message to Armenia, Azerbaijan, and Georgia that might be paraphrased as "do not look a gift horse in the mouth."
Instead of focusing on membership, he says, the three should take advantage of what the EU is already offering -- including a special "neighborhood" policy that could pave the way for more investment and economic opportunity.
"We must give [the three South Caucasus countries] this message: 'Europe is a big market. In the future, when you're building your economy, you can export to Europe -- and indeed not only energy. Europe is [also] a big investor. It is enormous...the biggest economic [actor] in the world and [it] is not far from the three countries.' So, we are ready to invest. We must prepare a climate in the three countries. And then in our aid we always help to build infrastructure linking the three countries or helping [train] customs officials to facilitate trade and the movement of people, etc.," Prodi said.
Concerning eventual EU membership, Prodi says this is not possible as long as the borders of Europe remain unclear. Prodi acknowledges the EU's basic treaties promise membership to all "European" countries. But, he says, Europe's borders are not fixed and change with time.
He adds that now -- just after the EU has added some 10 new members -- is a bad time to fix the borders. He says the European public must be sent a message that Europe is not enlarging every day.
"But this does not mean that these three countries do not belong to Europe. They don't belong now, and it is not planned that they belong in the 'European Europe.' But the doctrine of the neighborhood policy that we worked on so long and so deeply is [there] to build links of friendship and cooperation, strong links with countries, which for the foreseeable are not members of the [European] Union," Prodi said.
Prodi does not exclude the possibility that the border of Europe may in 2015 run between Turkey on the one hand and Georgia and Armenia on the other. He hastens to caution that the EU has yet to decide whether to open accession talks with Turkey -- the commission is due to make its recommendation on 6 October. But he points out that once talks begin, they do so "with the perspective of closing them."
Prodi also directs his comments to Russia. He says Russia is a key partner and an important player when it comes to the future of the South Caucasus.
He avoids comment on recent Russian statements indicating the possibility of pre-emptive strikes on terrorist targets outside its own borders. But he does say that Russia, in his opinion, is not interested in destabilizing the region.
"I think that now Russia is interested in promoting stability and security in the area. This is what I think, and I know that the Russians are wise, they have no interest to enlarge any conflict," Prodi said.
Prodi suggests that the EU -- as a "strong, independent" friend of Russia -- can help in resolving what he calls the "frozen conflicts" of the South Caucasus. But he rules out any EU military presence in the region, at least for the time being.
Prodi says force is not an option for any of the problems of any of the three countries. This is particularly true of Georgia and its troubles with North Ossetia and Abkhazia. Prodi refuses to directly indicate whether Tbilisi's use of massive force would jeopardize its blossoming ties with the EU.
He also observes that Georgia has in recent history used "intelligent force" -- this is a veiled reference to the largely peacefully resolved conflict with Adjara.
Prodi says the EU engages in no favoritism among the three countries, although Georgia is, so far, alone among the three to have benefited from an international donors' conference which netted a many million dollar windfall.
Prodi promises the EU will be an "honest broker" among the three countries.
"Globally, Georgia receives more money. But if you analyze European policy, it has been very honest between the three countries. I am not happy that the amount of resources has not been enough for the terribly big needs of the countries, but I want to have a balanced strategy," Prodi said.
He says no donor conferences are planned for Armenia or Azerbaijan at this stage.