The result means that the draft charter for a post-Saddam Hussein Iraq could be defeated if one more province musters a majority of two-thirds against it, under election rules.
Officials announced on Saturday that the province of Salah al-Din had rejected the constitution by 81% in the 15 October vote.
If three of Iraq's 18 provinces vote against the charter with a two-thirds majority, it will be rejected, new elections held and a new draft written.
The results of a final four provinces - Arbil, Babel, Basra and Nineveh - should be available on Tuesday or Wednesday, senior electoral official Abdul Husain al-Hindawi said on Monday.
While the first three are expected to easily approve the constitution, Sunni-dominated Nineveh, the capital of the mixed governorate of Mosul, looked set to decide the issue for the entire country.