The Sunnis object to several parts of the draft text agreed by Shia and Kurdish parties. But Shia leaders have said they will compromise no further.
It has emerged that US President George Bush phoned a Shia leader earlier this week urging him to seek consensus.
Mr Bush spoke to Abdul Aziz al-Hakim and "asked him to be more flexible with regard to Sunni demands," sources close to the ruling Shia alliance told Reuters.
'Final offer'
The Sunni marchers in Baquba danced and sang chants glorying Saddam Hussein, the deposed Iraqi leader who held onto power through a series of bloody crackdowns on the country's Shia and Kurdish communities.
"Bush, Bush, listen well: We all love Saddam Hussein!" sang some of the marchers, according to Reuters news agency.
The protesters also accused the US-backed government in Baghdad of promoting the interests of Iraq's Shia neighbour, Iran.
Placards in the procession proclaimed opposition to provisions in the constitution that the protesters claim would lead to the federalisation - and ultimate disintegration - of Iraq.
Sunni leaders have rejected the constitutional text drawn up by the Kurdish and Shia groups who hold a big majority of seats in the Iraqi parliament.
The outstanding issues from the Shia-Kurdish draft include: federalism, and the way to form semi-autonomous regions the terminology used in eradicating the influence of the former Baath regime - whether to use the term Baath party or Saddam's Baath structuring of authority between the presidency, parliament and the government.
Monday's deadline for agreement was extended until Thursday - and passed without a breakthrough.
Talks with Sunni leaders are continuing, although on Friday a top Shia official, Abbas al-Bayati, said Shias had made their "final proposals", including concessions on both federalism and the Baath Party.
He would not go into detail, but said "we cannot offer more than that".
Friday also saw Shias rallying in cities across southern Iraq to show their support for the radical cleric Moqtada Sadr and to demand the government improve public services.
Sunni isolation
The speaker of Iraq's parliament, Hajim al-Hassani, said whatever the outcome, the text would go to a referendum in October.
Shias and Kurds could approve the document in parliament without Sunni backing.
But the insurgency rocking Iraq has its roots in the Sunni heartlands and the constitution is supposed to be part of the process of winning the Sunni community round, the BBC's Baghdad correspondent Mike Wooldridge says.
Sunnis have expressed concerns that allowing for federalism may lead to the creation of an autonomous Shia area in southern Iraq - like the Kurdish north but under Iran's influence.
The Sunnis fear greater autonomy for the Kurdish north and Shia south could compromise their share of revenues from those oil-rich regions.
Sunni negotiator Saleh al-Mutlaq said one suggestion of delaying any new federal regions for two years was unacceptable to them and at the moment he believed there was no prospect of consensus.
The US has announced it is sending another 1,500 troops to Iraq to bolster security before the referendum on the constitution, due in October.
In that vote, Sunnis could block the constitution by delivering a two-thirds "No" vote in three of Iraq's 18 provinces.
That would mean parliament's dissolution and fresh elections in December.
Story from BBC NEWS: http://news.bbc.co.uk/go/pr/fr/-/1/hi/world/middle_east/4188112.stm
Thousands of Sunni Muslims have demonstrated in the Iraqi city of Baquba to protest against the draft constitution being debated in Baghdad.