ISTANBUL — With Turkey’s prospects for joining the European Union growing more elusive and the country reaching out to predominantly Muslim countries with a vigor not seen in years, a longstanding question is vexing the United States and Europe: Is this large, secular Muslim country turning East instead of West?

When President Obama visited Turkey in April — a symbolic gesture that underlined Turkey’s geostrategic importance — he emphasized the country’s role as a bridge between East and West, acknowledged its mediation in the Arab-Israeli conflict and threw his weight solidly behind Turkey becoming a European Union member.

Now, six months later, some in Washington and Brussels are questioning Turkey’s dependability as an ally, and many Turks are asking whether they should reject the European Union before the bloc rejects them.

Fears that Turkey is abandoning its bridge-building role were fanned this month when it canceled air force exercises with Israel, straining ties that frayed in January when Prime Minister Recep Tayyip Erdogan castigated Israel’s president, Shimon Peres, over the war in Gaza, in front of world leaders at Davos, Switzerland.

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