Either would make it the deadliest month since 2008.
The wave of violence comes about 18 months after the last American troops left Iraq, which is neither secure nor stable more than a decade after the US-led invasion of the country.
An AFP count, based on security and medical sources, put May's toll at 614 people killed and 1,550 wounded, while data from government ministries showed 681 had died and 1,097 were wounded.
The UN gave a significantly higher toll of 1,045 killed and 2,397 wounded.
On Saturday evening, top political and religious figures came together for a gathering that has been called for since late 2011, but continually delayed.
Prime Minister Nuri al-Maliki, a Shiite, embraced parliament speaker Osama al-Nujaifi, his main Sunni political rival, in a move symbolising an aim to reduce tensions in the country.
But no tangible measures were announced, and more meetings are to be held later.
"The situation in Iraq was very difficult, and on the edge of collapsing," Nujaifi told journalists, adding that the country on the verge of "civil war."
"This meeting was very important, and we must stabilise the situation," he said.
Meanwhile, President Jalal Talabani's doctor said the health of the veteran Kurdish leader, who has been receiving treatment for a stroke in Germany since December, was improving and he would be able to fulfil his official duties upon his return to the country.
The 79-year-old's condition has major political implications, as Talabani has played a prominent role in seeking reconciliation between Sunni and Shiite Muslims, and Kurds and Arabs.
Long-running disputes between top politicians have paralysed Iraq's government, and been linked by analysts and officials to increases in violence.
UN envoy Martin Kobler has warned that "systemic violence is ready to explode at any moment" if they do not resolve their disagreements.
While violence mainly targeted the government and members of the Shiite majority in the past, unrest in May was more wide-ranging. There were also major attacks on Sunnis, striking at all aspects of daily life.
Bombings cut down worshippers in mosques, shoppers in markets and people mourning those killed.
One Baghdad car bomb even tore through a group of people cheering a bride ahead of her wedding.
Violence has fallen from its peak at the height of the sectarian conflict in 2006 and 2007, when the monthly death toll repeatedly topped 1,000, but the death toll has begun to rise again.
There has been a heightened level of violence since the beginning of the year, coinciding with rising discontent in the Sunni Arab minority that erupted into protests in late December.
Sunnis, who ruled Iraq from its establishment after World War I until Saddam Hussein's overthrow in 2003, accuse the Shiite-led government of marginalising and targeting them.
Analysts say government policies that have disenfranchised Sunnis have given militant groups both fuel and room to manoeuvre among the disillusioned community.
The government has made some concessions to placate protesters and Sunnis in general, including freeing prisoners and raising the salaries of Sunni anti-Al-Qaeda fighters, but underlying issues have yet to be addressed.
Analysts say dealing with political issues, including Sunni grievances and disputes over issues ranging from control of territory to power-sharing, is key to curbing the violence.
"The government should genuinely (take) steps toward the negotiation with the street, with the (Sunni) protesters," said Maria Fantappie, an analyst with the International Crisis Group.
"So far, the government has not taken any genuine steps towards really... engaging into a dialogue with the protesters," she said.