Iraqi member of Parliament Humam Hamoudi called Parliament Speaker Salim Jabouri's meeting with Barzani "disappointing" and "unfortunate" and said Jabouri went to Barzani in a personal capacity, not as Parliament's representative.
Two days of high level visits by Baghdad politicians to Irbil, the seat of the Kurdish regional government, have failed to resolve the impasse between Baghdad and its Kurdish region, which voted for independence in a non-binding referendum two weeks ago.
Jabouri's visit, intended to break the deadlock, instead underscored the divisions within the capital over how to respond to the Kurdish vote.
Few if any of Iraq's national politicians want to see an independent Kurdistan, but there is little consensus beyond that.
Shiite politicians, especially those close to Iran, have urged a hard line against the Kurdish region. They see an opportunity to clip Barzani's wings, which they see as the biggest obstacle to expanded Iranian influence in north Iraq, said Hadi Meraie, a political analyst and head of the Iraq Observatory for Press Freedoms.
"The Shiite groups consider that Barzani has crossed a red line to threaten Iraq's unity and by extension Iran's influence over the region," said Meraie.
The war on the Islamic State group has drawn the country's Shiite-dominated Popular Mobilization Front militias, a main vessel of Iranian influence, deep into Sunni-majority northern Iraq, raising tensions with Kurdish Peshmerga militias also invested in the fight.
Sunnis, including Parliament Speaker Jabouri, and some Shiite politicians prefer to see Iran's influence contained.
Hamoudi is a leading member of the Iran-backed National Iraqi Alliance block, which controls the majority of seats in Parliament.
Last month's referendum opened the door for Baghdad to reclaim oil-rich Kirkuk, an ethnically-mixed city outside the Kurdish zone that is currently under Peshmerga administration. The militia was able to claim the city after national forces fled from an IS advance in 2014.
The PMF, already distributed near the city, would be the big winners of a re-nationalized Kirkuk.
"The crisis is at still in its formative stages, not in its resolution," cautioned Meraie.
Iraq's Kurdish region voted for independence in a symbolic but controversial referendum two weeks ago. Baghdad responded by banning international flights out of the region and threatening to suspend Kurdish representatives from the national parliament. The vote was largely opposed by the region's Arabs and other minorities.
Prime Minister Haidar al-Abadi demanded the Kurdish self-government annul the results and called for joint administration over Kirkuk.
Turkey and Iran also threatened punitive measures against the Kurdish region, fearing Kurds in their own countries would renew their campaigns for self-rule.
Jabouri's meeting with Barzani came a day after Parliament adjourned itself to allow more time to resolve the crisis.
"We met to stop the deterioration of relations between the center and the region," said Jabouri in a statement after the meeting. He said the two leaders discussed Kirkuk.
Last week's passing of former President Jalal Talabani, a Kurdish leader regarded as a unifying figure in post-invasion Iraq, failed to reconcile the two sides. Abadi skipped the funeral Thursday, held in the Kurdish city of Suleimaniyah. Talabani's casket was draped in a Kurdish flag.
Barzani's office said he and two of Iraq's three vice presidents agreed Saturday to restore relations with Baghdad after a meeting in Irbil. The vice presidents' offices denied any resolution.
Iraq's landlocked Kurdish region produces up to a quarter of Iraq's petroleum output.
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