Pro-coup demonstrators in Cairo’s Tahrir Square were given a tangible reminder of America’s long-standing support for Egypt’s military last month, when a missile-equipped, U.S.-made Apache helicopter flew low overhead and dropped tiny Egyptian flags to the cheering crowd.

Helicopters may soon become another kind of symbol, this time of decreasing U.S. support, as the Obama administration debates whether to stop next month’s scheduled delivery of new Apache AH-64D aircraft, according to a senior U.S. official.

Refusal to send the Apaches, part of an $820 million, 12-aircraft order dating from 2009, would fall far short of the suspension of all U.S. military aid to Egypt — including the crucial spare parts to maintain the American-made equipment — that some have demanded.

There is little indication that stopping the helicopter shipment would have any more effect on the Egyptian military’s conduct than last month’s suspension of U.S. F-16 fighter jet deliveries, or President Obama’s cancellation of a joint military exercise in the wake of the past week’s brutal crackdown on anti-coup protesters that left more than 600 dead. Egypt has about three dozen Apaches from previous aid packages.

For now, the Obama administration is playing a balancing game, trying to send tactically sharp messages while preserving influence in an increasingly polarized society, protecting other national security interests in the region and positioning the United States for a long-term strategic relationship.

Obama said Thursday that he has ordered his national security team to “assess the implications of actions taken” by Egypt’s current rulers and of “further steps that we may take as necessary.”

Cancellation of the helicopter shipments would be the first response to that order. Payments on other aid-financed military supplies for Egypt — largely owed to U.S. defense contractors — will soon be due. If the Egyptian military does not figure out a way to stop the violence and move expeditiously along a path to restore civilian government, more cuts may follow.

Many lawmakers back Obama’s cautious approach. So do Israel and powerful Persian Gulf nations that oppose ousted president Mohamed Morsi’s Muslim Brotherhood and are willing to bail out Egypt’s drowning economy.

But even before Wednesday’s rampage by security forces in Cairo, the U.S. administration’s incremental response to the military’s July 3 overthrow and arrest of the democratically elected Morsi had left Obama vulnerable to accusations of weakness and moral vacillation. Some said he was ensuring the long-term instability he has said he is trying to stem in Egypt.

“The failure of the Obama administration to use our influence to shape events in this critical part of the world has only diminished our credibility, limited our influence, and constrained our policy options,” Sens. John McCain (R-Ariz.) and Lindsey O. Graham (R-S.C.) said in a statement Friday. The two are part of a growing bipartisan chorus demanding an end to nearly $1.6 billion in annual U.S. assistance to Egypt, $1.3 billion of it used to finance purchases of American military equipment.

2009 speech in Cairo

The relative lack of U.S. engagement as crises roil Syria and Egypt may produce the very thing Obama eschewed in his seminal speech to the Islamic world, delivered in Cairo in 2009, said Tamara Cofman Wittes, Obama’s former deputy assistant secretary of state for Near Eastern affairs and now director of the Saban Center for Middle East Policy at the Brookings Institution.

In the speech, Obama promised a “new beginning” in America’s relationship with Muslims, based on “mutual interest and mutual respect” rather than the military might evidenced in Iraq.

Yet as Syria falls into sectarian civil war and Egypt’s secular and Islamist divide deepens, Wittes said, “we may indeed see the emergence of a much wider, fiercer threat of violence to the Arab world that will demand a certain type of American engagement that is precisely the opposite of what the president was aiming for.”

Thomas Carothers, vice president of studies at the Carnegie Endowment for International Peace, cited a different Obama speech, delivered at the State Department in May 2011 in the midst of the NATO air offensive in Libya. Speaking of the unfolding Arab Spring, Obama said the United States would stand firm in opposition to “the use of violence and repression” in the region.

“There will be times when our short-term interests don’t align perfectly with our long-term vision for the region,” Obama said. “But we can, and we will, speak out for a set of core principles.”

“He was saying ‘get on the right side of history,’ ” Carothers said. “But he didn’t really follow through” in the half of the Arab world — including close U.S. allies such as Bahrain and Saudi Arabia — that remains undemocratic and often repressive.

In Egypt, Carothers said, stability and the basic “caution of the president” have repeatedly prevailed over values-based ­risk-taking. “U.S. policy toward Egypt has been stuck in deep relationship management mode, whose purpose is to maintain a relationship with whoever’s in power,” he said. Since the coup, “the military has reached the conclusion they can do anything and they will maintain a relationship with the U.S. government. And so far they’re right.”

Advice went unheeded

Many critics fault the administration as much for what they see as its failure to take a strong stand against abuses by Morsi’s government as they do for its approach to the Egyptian military.

Several U.S. officials, who spoke on the condition that they not be identified or quoted in discussing the ongoing Egyptian crisis, took strong issue with that criticism. While only Egyptians can work out their differences, they said, private U.S. interventions with Morsi were pointed and robust.

Specifically, the administration repeatedly advised Morsi to reach out to the largely secular opposition that spearheaded the early 2011 nationwide upheaval that forced Hosni Mubarak’s departure after three decades in power. And it urged Morsi to accept an internationally brokered economic rescue from the International Monetary Fund.

As opposition protests grew and the economy fell apart, increasingly urgent, high-level U.S. appeals fell on deaf ears, even as the opposition to Morsi accused the United States of failing to restrain him.

On June 30, the anniversary of Morsi’s first year in office, millions of Egyptians took to the streets to call for his resignation. In less than a week, the military had arrested him and other senior Muslim Brotherhood officials and taken power.

The administration resisted calling the ouster a coup — a determination that would trigger an automatic aid cutoff under U.S. law — and noted that millions of Egyptians applauded the military action. Secretary of State John F. Kerry referred to the military action as “restoring democracy,” and Deputy Secretary of State William J. Burns, on a visit to Cairo in mid-July, called it “a moment when Egyptians have a second chance to put their post-revolutionary transition on a successful path.”

Today, as the military’s declared road map for return to elected civilian rule has been eclipsed by violence and armed forces commander Gen. Abdel Fatah al-Sissi has become the country’ s de facto leader, U.S. officials are making the same appeal to the military: Include the other side in building the new Egypt.

Talk of a “second chance” at revolution has infuriated even some of the administration’s closest friends. In a reference to the unambiguous wording of the U.S. anti-coup law, Sen. Patrick J. Leahy (D-Vt.), who chairs the Appropriations subcommittee in charge of U.S. foreign aid, said after Obama’s Thursday statement that “aid to the Egyptian military should cease unless they restore democracy.”

Ernesto Londoño contributed to this report.