, as President Hassan Rouhani of Iran declared readiness to defend Iraq’s Shiite holy sites with force and Saudi leaders issued a barely veiled admonishment not to intervene, the New York Times said in news briefed by “Shafaq News”.

The sharpened tone coming from both countries, which have long regarded each other with suspicion — Iran is overwhelmingly Shiite and Saudi Arabia Sunni — suggested that their recent tentative efforts to improve relations might be faltering over the Iraq crisis.

Both countries have long vied for influence in the region and support opposing sides in Syria’s civil war.

The Saudis are also increasingly concerned about Iran’s efforts to ease its longstanding estrangement with the United States, a close strategic and economic partner of Saudi Arabia. While the United States and Iran differ on many issues, most notably Iran’s nuclear program and the Syria conflict, they see a shared interest in arresting Iraq’s Sunni insurgency, The newspaper added.

In a televised speech in Iran on Wednesday, Mr. Rouhani expressed full support for the Iraqis who have joined volunteer militias to fight the Islamic State in Iraq and Syria, the Sunni extremist group that straddles both countries. It has seized large sections of northern and western Iraq over the past few weeks, threatened to march on Baghdad and vowed to obliterate cities and shrines dear to Shiites.

“Regarding the holy Shiite shrines in Karbala, Najaf, Kadhimiya and Samarra, we announce to the killers and terrorists that the big Iranian nation will not hesitate to protect holy shrines,” the president vowed in the speech to a crowd in Lorestan Province in western Iran. “These terrorist groups, and those that fund them, both in the region and in the international arena, are nothing, and hopefully they will be put in their own place.”

Mr. Rouhani also said many Iranian volunteers were prepared to travel to Iraq to defend religious sites. He sought to cast them as the allies of patriotic Iraqis from all backgrounds who see the Sunni insurgents as a scourge — a theme he also emphasized in a posting on his

“Iranian nation will protect Iraq’s holy shrines & they aren’t alone. Iraq’s Sunnis, Shias & Kurds all ready to defeat terrorism solidarity,” he wrote.

Mr. Rouhani signaled over the weekend that Iran did not intend to send troops to Iraq. But Gen. Qassim Suleimani, the head of Iran’s paramilitary Quds Force of the Revolutionary Guards, recently traveled to Iraq to meet with Iraqi leaders, who have mobilized thousands of militia fighters, almost exclusively Shiites. The high-level contact suggested that General Suleimani was helping oversee their training and strategy.

In Saudi Arabia, Foreign Minister Prince Saud al-Faisal warned that no outside power should be meddling in Iraq. Coming on the same day that Mr. Rouhani spoke, the clear implication was that the Saudi minister was referring to the Iranians. He said the Iraqis needed to achieve national reconciliation without “foreign interference or outside agendas.”

The Saudi kingdom also issued a statement repudiating accusations by Iraq’s Iranian-backed Shiite prime minister, Nuri Kamal al-Maliki, that it was providing moral or financial support to the Sunni insurgents.

“Any suggestion to the contrary is a malicious falsehood,” the Saudi statement said.

Both Iran and Saudi Arabia had taken steps to ease their strained ties in recent months, helped in part by the election last year of Mr. Rouhani, who said he wanted improved relationships with Arab neighbors.

In January, Iran’s foreign minister, Mohammed Javad Zarif, in a visit to Lebanon, played down suggestions that Saudi operatives had bombed Iran’s embassy there last November and instead emphasized Iran’s desire for better relations in the region — especially Saudi Arabia. In May, the Saudi foreign minister said he had invited Mr. Zarif to visit, the New York Times concluded.