Four F-16 fighter jets were scheduled to fly to Egypt on Tuesday morning as part of a U.S. military aid package worth more than $1 billion a year -- but the shipment has run into delays over apparent "political" issues.

If the Obama administration is able to send the planes, it will mark the first known military aid to Egypt since millions of Egyptians protested the rule of Mohammed Morsi, leading the Egyptian military to remove him from power earlier this month.

Supporters say that such aid is critical because it gives the U.S. influence over the Egyptian military. But critics say it is a waste of money, or worse -- a gift of weapons that could later be turned against American interests.

The shipment has now been delayed at least 24 hours due to "political reasons," according to a source who works on the naval air base in Fort Worth, Texas, from where the planes were being sent.

Officials at the U.S. Department of State, asked by FoxNews.com about the unexpected delay, explained that "we are reviewing our obligations and are consulting with Congress about the way forward."

The statement refers to the legality of sending the planes to Egypt, which is questionable because of a 2012 law that forbids the president from sending military aid to any government that has come to power in a "military coup."

So far, the Obama administration has carefully avoided using the phrase "military coup" to describe the overthrow of Morsi's government in Egypt. On July 10, White House spokesman Jay Carney defended continued military aid.

"We do not believe it is in the best interest of the United States to make immediate changes to our assistance programs," he said.

But that attitude sparked criticism from some, including Sen. Rand Paul, R-Ky.

"By the President's refusal to call the situation in Egypt a 'coup' and continuing the flow of foreign assistance to Egypt, he is forthrightly saying 'I am ignoring the rule of law,'" Paul said in a statement.

Paul has introduced a bill titled the Egyptian Military Coup Act to officially declare that a coup took place and halt the shipments of aid to Egypt, but it is considered unlikely to pass.

Morsi was democratically elected, but at one point unsuccessfully attempted to seize dictatorial powers and also pushed through a new constitution based more strictly on Islamic law. Before coming to power, Morsi led the Muslim Brotherhood in Egypt, and urged that "we must not forget to nurse our children and grandchildren on hatred towards those Zionists and Jews, and all those who support them."

While Morsi was in charge, the Obama administration gave 12 F-16 fighter jets to the Egyptian military, and some criticized it for supporting Morsi's government. But the military overthrow of Morsi now casts those shipments in a different light, some security experts said, and highlights the importance of having influence over the Egyptian military.

"If we want to have as much influence as possible over the military, and we want them to push for a real constitution, the best thing we can do is make it clear we are going to continue the arms deliveries," said Tony Anthony H. Cordesman, who has served as a consultant for the State and Defense departments and who holds the Burke Chair in Strategy at the Center for Strategic and International Studies.

Previous statements from officials at the State Department to FoxNews.com in April also defended the aid by hinting at differences of opinion between the Egyptian military and Morsi's government.

"The Egyptian military has long had cordial ties with Israel and is a pillar of support for the peace treaty within the Egyptian Government," one said.

"For the past 30 years, the F-16 aircraft has been a key component of the relationship between the United States military and the Egyptian Armed Forces," the statement continued.

But some say that such gifts to the Egyptian military are a bad idea.

"We're sending U.S. military aid to, in essence, a military-run government ... one that just backslid from democracy in a military coup," said Malou Innocent, a foreign policy expert at the Cato Institute.

"The question is, are we getting enough for our aid in terms of the democratic accountability? And there is very little indication that the Egyptian military would not have done what it did anyway -- independent of US aid," she said.

Innocent said that taxpayers wouldn't be happy if they knew how this money was spent.

"If you ask the average American taxpayer: 'what would you prefer to do with those billions of dollars?' they would prefer to keep it at home ... rather than give it to the Egyptian military, which is continuing to trample on Egyptian liberties."

Cordesman, however, argues that having a moderate government in Egypt is more important than having a democratically elected one.

"An open-ended support for democracy, regardless of who wins, may not always be the wisest course of action ... a good election with a bad government is a disaster. And bad election with a good government is a success," he said.

He added that the cost of the aid is unfortunate but necessary.

"In an ideal world would we send the money -- of course not. In a practical world, the cost of not having close ties with the military, both in terms of Egypt's future and regional stability, is greater."

"If you cut the arms shipments off, you make it an issue in terms of the nationalism, prestige, and the honor of the Egyptian military -- and instead of having leverage, you leave yourself without any clear influence."

Innocent said that cutting aid might actually send a positive signal -- that the U.S. would not tolerate military coups -- but that military aid should at least be gradually halted.

"We could phase aid out over a three-to-five year period so it's not ripping off the band-aid all at once," she said.

"We need to be asking those broader questions, like: Do we want to promote democracy or not? ... What we have now is foreign policy on auto-pilot."

The author of the story can be reached at maxim.lott@foxnews.com or on Twitter at @maximlott