Talabani, a Kurd, on Friday accused al-Jaafari, a Shia, of unilaterally taking decisions in violation of accords signed by the two blocs before the setting up of the government.
"One of our problems with the prime minister is that he violates the law. For instance, on the concept of government and the cabinet, the state administration law for the interim period has clearly defined everything," Talabani said.
"As based on Article 24 of the state administration law, the government is composed of the national assembly leadership, the cabinet and the judiciary. This means the cabinet is not [solely] the government."
Talabani added that the cabinet cannot replace the government or exercise its duties exclusively, since it is only part of the government.
"The cabinet cannot alone assume the executive power, which is designated to the presidency of the republic and the council of ministers," Talabani said.
The president also highlighted the differences between al-Jaafari and the Kurdish alliance.
He said the Kurdish alliance had submitted a memo to al-Jaafari for implementing the joint agreements on which the coalition between the Kurdish and the Iraqi alliances was based.
Last month a row erupted between Talabani and al-Jaafari over who should lead Iraq's delegation to the annual UN summit in New York.
According to officials, the dispute began when Talabani sent a letter to al-Jaafari declaring that as Iraq's titular head of state, he would be representing the country at the annual UN session.
Al-Jaafari was unable to represent
Iraq at the UN General Assembly
Al-Jaafari replied by saying that since he runs the government, he should lead the delegation. The prime minister suggested that both he and Talabani could deliver General Assembly speeches, but the president refused.
A Shia politician who refused to be identified said al-Jaafari decided that unless he could make a speech too, he would stay home.
Writer and journalist Nazar al-Samirraei told Aljazeera the differences were expected.
"Since assuming the presidency, it was widely believed Jalal Talabani should not only be a nominal president for protocol, as he has rich political experience and supported by a strong alliance. This is why he was expected to spark these differences.
"But what is so remarkable is that the dispute has arisen just two weeks ahead of the referendum on the Iraqi constitution and three days after a three-man meeting at Salah al-Din resort between Talabani, Barzani and US ambassador Khalilzad Zalmay."
He added that this indicated Talabani was no longer willing to tolerate sharing the consequences of the mistakes made by al-Jaafari's government.
However, al-Samirraei added that "the joint agreement signed between the two alliances to which Talabani referred still remains a mystery".
Separation of powers
Under Iraq's interim constitution, Talabani leads a three-member council that serves collectively as head of state.
But al-Jaafari leads the government, similar to Germany which has a president with few powers and a chancellor who governs.
In May, Talabani represented Iraq in the Summit of South American-Arab in Brazil, which took place less than two weeks after al-Jaafari's government took power.
The differences come as Shia and Kurds are trying to maintain unity to promote the new constitution in a 15 October referendum.