Adeeb, who is a candidate for the city of Karbala, said that Iraqiya's progress during the 10 days of vote counting had been "like a miracle".
However, the early results do not necessarily mean that Iraqiya is winning the race for the most seats in parliament.
State of Law leads in Baghdad, which accounts for one-fifth of Iraq's 325-member Council of Representatives, as well as six other provinces, with Iraqiya leading in five of the 18.
Al Jazeera's Anita McNaught, reporting from Baghdad, cautioned that the current tally includes only the general vote, which is for Iraqis voting inside Iraq.
"We still have two other big significant voting blocs to be included in the tally," she said.
The special vote, which was conducted on March 4 for military personnel, hospital patients, handicapped and prisoners, and the expatriate vote are still to be counted.
The country's proportional representation system makes it unlikely that any single group will clinch the 163 seats needed to form a government on its own, and protracted coalition building is likely.
The counting process has been fraught with claims of fraud, mostly from the opposition.
Al-Maliki himself has signed a letter to Iraq's Independent High Electoral Commission, stating that State of Law has "received reliable information that supervisors of the electronic counting centre" are linked to rivals contesting the poll, including some supporting Allawi.
It said the political allegiance of the counting centre's supervisors undermines "their neutrality in administering such a momentous and crucial process", The Associated Press news agency reported.
The letter calls for an investigation into the political ties of all officials and employees at the counting centre and argues that final results should not be released until all complaints are investigated "however long it takes".
Haidar al-Mulla, an Iraqiya spokesman, criticised al-Maliki's accusations.
"When we said there were some violations, he labelled our calls as losers' allegations. Now we proved to be at the lead he is complaining," al-Mulla told Al Jazeera.
"Even if we win, we will still investigate the violations, and we will send them to the courts. We will not be like him, taking the law by our own hands. We will even fight to get him or any other rival every right he or she is entitled for. That is why people are voting for us."
Michael Hanna, an analyst on Iraqi affairs at the Century Foundation in New York said: "Because it is so tight, it is more tense and you are going to see more allegations of fraud. People could try to use that as a political weapon."
He said al-Maliki's allegations could reflect the fact that the prime minister's coalition has realised their lead is not as strong as once believed.
"It is hard not to be cynical about some of these claims, most of them actually."
Electoral commission official Saad al-Rawi confirmed the commission had received al-Maliki's complaint but said it was one of many to come in without concrete evidence.
He acknowledged that six workers at the counting centre had been fired, but said it was for incompetence and entering incorrect data into the computers.
Independent Iraqi observers and UN officials advising the commission say they have seen no evidence of widespread fraud that could undermine the outcome.
Once the electoral commission announces the final poll results, the country's supreme court will have to certify them - after hearing appeals - within about a month of the election.