Since 16 June, panoply of US operations in Iraq have expanded to include around 60 surveillance flights over Isis territory

America’s newest war in Iraq has cost over half a billion dollars so far, according to Pentagon estimates, all before President Barack Obama decides upon a strategy against Islamic State (Isis) militants.

Rear Adm John Kirby, the Pentagon press secretary, told reporters on Friday that daily military operations in Iraq since 16 June, when the White House informed Congress it had ordered up to 275 US troops to bolster embassy security in Baghdad, have cost on average $7.5m.

Those operations stretched into their 75th day on Friday, suggesting a cost of around $562.5m. Kirby did not break down the $7.5m figure, and he cautioned that not that much money was actually spent every day.

Since 16 June, the panoply of US operations in Iraq, the first since the 2011 troop withdrawal, have expanded. Hundreds of US special operations “advisers” arrived in Baghdad and Irbil to aid the Iraqi military and Kurdish Peshmerga militia forces plan a response to Isis’ advance across much of northern and central Iraq, briefly topping 1,000 for a reconnaissance mission atop Mount Sinjar earlier this month.

Around 60 surveillance flights, by piloted planes and drones, occur daily over territory controlled by Isis. The US has launched 110 airstrikes against Isis positions, artillery and equipment, mostly to dislodge Isis from the Mosul Dam and prevent the jihadist army from retaking it. Initially, those airstrikes centered around relieving an Isis siege of Mount Sinjar and stopping Isis from advancing into Iraqi Kurdistan.

The most recent airstrikes occurred Friday, Central Command said, and took place near the dam. It said it destroyed four Isis “armed vehicles” and three more “support vehicles,” with another armed vehicle “severely damaged.”

Kirby suggested that the hundreds of special operations “advisers,” some of whom are said to spot for airstrikes, might at some point perform “more of an advisory mission” to bolster the performance of Iraqi brigades. Much of the advice currently offered occurs within senior-level planning centers, called Joint Operations Centers, rather than at frontline brigades.

Kirby also said that the Pentagon had yet to provide Obama with military options against Isis in Syria, where he is under enormous political pressure to expand the latest US war. Obama conceded on Thursday that he thus far lacks a strategy against a group that has erased the border between Iraq and Syria, and which the Pentagon leadership describes in apocalyptic terms.

“When we get to a point when we’re ready to have a more fulsome discussion about that, the Pentagon will be ready to have that discussion,” Kirby said.

For months, however, Pentagon planners have discussed attacking Isis in Syria as well as in Iraq. Last week, General Martin Dempsey, the chairman of the Joint Chiefs of Staff, told reporters he was “not predicting [airstrikes] will occur in Syria,” a tempering note echoed by Obama on Thursday.

Kirby conceded that he would “be less than truthful if I said to you that we hadn’t been thinking about that before yesterday. Of course we have been … but we’re not at the point where we’re prepared to have a more fulsome discussion about what those options are with the commander-in-chief.”

Military planning, Kirby said, is “an iterative process.”

The US’ closest ally, the UK, raised its terror level to “severe” in response to Isis on Friday, as western governments fear that an unknown number of Isis fighters holding US and European passports will return from Iraq and Syria to launch attacks. But the Obama administration cautioned that it sees no imminent domestic attack from Isis.

“At present, the US Department of Homeland Security and the FBI are unaware of any specific, credible threat to the US homeland from [Isis],” Homeland Security Secretary Jeh Johnson said in a statement.

“Plainly, however, violent extremists who support [Isis] have demonstrated the intent and capability to target American citizens overseas, and [Isis] constitutes an active and serious threat within the region.”