It is a question that goes right to the heart of the country's long-standing debate over identity. Is it part of the Muslim world and the Middle East, or of the West and Europe? Or, is it possible to keep all of these identities at the same time?
Prime Minister Recip Tayyip Erdogan seemed to suggest an answer last month, at the Jidda Economic Forum in Saudi Arabia.
"Whatever happens," he told delegates, "we will not base relations on ethnic and religious roots or geography. There is nothing like this in economic relations."
Erdogan's words were a rejection of suggestions that an Islamic common market should be formed. Meanwhile, Erdogan's government is pushing harder than its predecessors to get Turkey a berth in the EU.
"The government is doing all that is in its power to join the EU," says leading columnist and commentator Fehmi Koru. "But those in power are also Muslims, so a part of the Islamic world. There are many people who are uncomfortable with all this."
Erdogan's party, Justice and Development (AKP), comes from a long tradition of political Islam in Turkey. Since the 1960s, "Islamist" parties have taken part in elections in the 98% Muslim country, although strict secular laws have led to the repeated banning of such groups.
The last to be banned was the Virtue Party (FP), under whose flag Erdogan and most of the current AKP leadership were previously identified.
Former Turkish PM Necmettin
Erbakan is a Euro-sceptic
Before that, they were leading lights in the Welfare Party (RP), which was also banned after being ejected from office in the "soft coup" of 1997.
Welfare's leader, Necmettin Erbakan, had long been opposed to Turkish membership of the EU. One leading RP official had this to say in 1994:
"Our opposition to the EU is based on the idea that we are from a different culture, we have a different identity and a different economic structure from European countries."
His name was Abdullah Gul - now foreign minister and deputy prime minister in the AKP government. Gul has recently been a key player in promoting Turkey as an ideal EU candidate member.
"Essentially," says Professor Iltar Turan of Istanbul's Bilgi University, "what the current government is trying to do - and Erdogan was trying to do in Jidda - is to make the point that they are not engaged in Islamist politics."
Rejecting stronger links with the Islamic world while boosting those with Europe are ways of re-branding the party's image and policies. The hope seems to be that Turkey can become more democratic.
"Many feel we have to be members of the EU not because we are Europeans," says Koru, "but because this is the best way to democratise the country ... The EU membership criteria are more important than membership itself."
These criteria oblige the country to improve its human rights record and prioritise social and economic reform.
"The Islamic world is largely governed by dictators," says Koru. "People have few rights there, and that affects our thinking greatly."
"The policy of the AKP also shows an appreciation that Islam doesn't constitute enough of a reference point for bringing about an economic or political common market," agrees Turan. "Erdogan does not see Islam as a basis for world integration."
Attempts by the old RP government to turn towards the Islamic world also led to questions about Turkey's identity.
"Every time Turkey has tried to boost its contacts with the Middle East and the Islamic world, people become more aware that only sharing a religion doesn't count for so much," adds Turan.
Turkey's leaders do not see Islam
as a basis for global integration
"These days, it's unrealistic to think of Turkey as a Middle Eastern country. Turks don't see themselves that way, but as a European, Mediterranean country."
Meanwhile, there is another question.
"The thing to discuss is not Turkey's view of Islam," says columnist and commentator Nuray Mert, "but the Islamic countries' view of Turkey. From ordinary people, there is generally a sympathetic view; but the official political view is that Turkey has changed to the West's side."
"The problem is," says Mert, "we are not discussing what the EU is. The EU is not only about economy. There are things in it about family relations, parental rights - just remember the clash there has been between Italian constitutional law and the EU."
Joining Europe means taking on board many legal frameworks that might seem quite alien to Turks at present - and to Islam. Membership also means a partial transfer of political - and perhaps in the future, military - power.
"The issue still to be discussed about the EU and Turkey is the transfer of sovereignty," continues Mert. "There are two contradictory things together here; we do not want to transfer our sovereignty, but we want to enter EU."
Often, this area of debate is ignored - with the discussion quickly refocusing on how Europe does not want Turkey.
"The Europeans don't feel we are European," argues Koru. "If you write a European constitution which says Christian values are central, how can I feel European? If the French pass a law banning headscarves from high schools, how can I feel part of that?"
Turkey and Europe seem in many ways then an unlikely partnership. Yet partnership may also be of great value, precisely because of their contrary natures.
"If we became members, Europeans would become more aware of their Islamic past"
"If we became members," suggests Koru, "Europeans would become more aware of their Islamic past.
"For the past six or seven centuries, Islam has been part of Europe. If you look to the roots of the Renaissance, for example, you see many Islamic scholars and philosophers.
"I myself, my parents and grandparents were born in Europe. If they accepted us, they would find something very enriching - that Islam is part of their culture too."
Yet where this leaves Turkey's relations with its Islamic neighbours remains an awkward question.
"Turkey is a democratic Muslim country," says Mert. "In his recent speech to the nation, Erdogan mentioned 'the Islamic geography that Turkey is tied to with the ties of brotherhood.' I wonder how consciously he meant that?"