BEIRUT, Lebanon—When the minions of the self-anointed caliph in the self-declared Islamic State that now straddles Iraq and Syria blow up a mosque supposed to contain the remains of the prophet Jonah, or offer punctilious details about the kind of purdah to be imposed on women, the world takes brief notice. But the group’s military campaigns have made less news in recent weeks because they seemed to have stalled.

Now, according to Western military analysts, it’s time to start worrying again. Those studying the attacks by the group formerly known as ISIS see critical changes in the bombings and skirmishing by the caliph’s troops and their allies in and around Baghdad. Some experts warn that a blitzkrieg—a lightning attack—is imminent, and it will be one the beleaguered and squabbling politicians in the Iraqi capital are ill equipped to combat. But it is more likely to be a guerrilla and terrorist offensive than an all-out push along conventional military lines.

Until recently, Islamic militant action around Baghdad appeared sporadic, uncoordinated, and lacking a clear strategic purpose. But analysts at the U.S.-based think tank the Institute for the Study of War, who have been plotting the locations and types of attacks in the recent flurry of blasts buffeting the Iraqi capital, have noted a clear pattern developing. They say it suggests the Islamic State is building up to something big and is no longer just focused on consolidating its grip and developing governance in the lands it now controls.

The institute’s analysts predict the caliphate may be readying for an onslaught, possibly timed for the end of the holy month of Ramadan on Monday or during the Eid holiday celebrations this week. The aim would not be to seize Iraq’s capital, which has a very large Shia population with every incentive to fight to the death against an organization that slaughters Shia prisoner en masse. The purpose of the Islamic State offensive would be to sow mayhem and to keep Iraq’s state apparatus from recovering from its stunning defeats in June, when it lost control of Mosul, the second-largest city in the country.

A sustained bombing campaign could well finish off the government of the embattled Prime Minister Nuri al-Maliki, who is still trying to cling to power, despite indications that even Iran, his main foreign backer, thinks it is time for him to step down.

There has been a burst of attacks by bombers wearing suicide vests and also car blasts “along avenues of approach to the capital and also within Baghdad proper,” the institute notes in an intelligence update. Most significantly, the analysts say, IS has been launching coordinated suicide attacks like those on July 19 which involved half a dozen or more blasts in a day. IS has made these so-called wave attacks a signature feature in their terrorist repertoire but had not mounted one on the capital since May 13.

The July 19 blasts were spread across the city touching many points of the compass—and they were deep in predominantly Shia strongholds, including Kadimiyah in the northeast, Bayaa, Jihad and Saydiyah in the southwest and Abu Dashir in southern Baghdad, where the Iranian-backed Shia militia Asai’b Ahl al-Haq holds sway.

The unfolding attacks display the group’s “high level of inter-cell coordination, its reach into Baghdad proper, and its ability for multiple teams to communicate, even in the context of Baghdad’s heightened security posture since the fall of Mosul,” according to the institute’s assessment.

It is significant that IS seems to have no problem penetrating districts controlled by Shia militias, suggesting the group’s leadership may have infiltrated sleeper cells into the capital even before the fall of Mosul. The caliphate has maintained a drumbeat of attacks in the Shia stronghold of Sadr City despite heightened security there and the fact its location in northeastern Baghdad is on the other side of the city from the territory IS holds outside the capital. On July 16, for instance, jihadists detonated two car bombs in Sadr City in a simultaneous attack that left nine dead and dozens injured.

In the immediate wake of the IS capture of Mosul in early June, alarmed press reports suggested the Sunni militants were “creeping closer” to the capital. Even last month, some intelligence analysts in the region were arguing the jihadists would not launch a conventional offensive, but would instead focus on demoralizing commando raids and a suicide-bombing blitz in Baghdad to force the Iraqi state to defend its capital rather than mount an offensive to try to retake Mosul or other territory captured by IS.

Now the time for the bombing blitz and commando raids appears to be approaching. “There are clear signs that what they are doing now is to test Baghdad’s defenses and to gauge the reactions from Shia militias and the Iraqi army,” says a senior U.S. intelligence official based in the region. The information the jihadists glean from these operations can help them formulate specific attack plans.

What impresses the official, and other analysts in the region, isn’t just how expert and disciplined the jihadists are being in their approach to Baghdad, but they are doing this at the same time they are consolidating their hold on towns they have seized elsewhere—and they have launched a major offensive against the forces of Syria’s President Bashar al-Assad.

Last week, the caliphate seized another major oilfield in Syria. On Thursday the group’s fighters targeted Syrian army bases outside Raqqa and in Hasaka and Deir el Zour. IS celebrated the assaults by posting online photos of headless bodies that the group claimed were soldiers killed in the attacks.