Correspondents say fighting intensified on Thursday morning with both sides firing heavily at each other's positions.
Describing the fighting as "very serious," India has called on Bangladesh to pull back its troops.
Thousands of villagers on both sides of the border have fled the fighting and sought refuge in temporary camps and mountain slopes.
But fresh skirmishes were being reported from a disputed river island separating the state of Tripura from Bangladesh.
Officials in Bangladesh's Kurigram district said India's Border Security Force used mortars and heavy machineguns.
The Bangladesh side fired machineguns and other weapons.
But they denied Indian media reports that said Bangladesh had fired rockets, injuring nine Indian civilians.
Indian warning
In the Indian parliament, Foreign Minister Jaswant Singh said the priority was for Bangladesh to withdraw from territory it seized along the border.
"We have very clearly told the Bangladesh Government about our views that there should be an earliest restoration of the status quo," Mr Singh said.
Correspondents say fighting intensified on Thursday morning with both sides firing heavily at each other's positions.
"There was continuous firing overnight and people are terrified. At least three artillery mortar shells fired by Bangladesh soldiers fell in towns and villages close to the border," said district police chief AJ Baruah.
"Hundreds of women and children are moving out of their villages for fear of being caught in the crossfire."
A senior official of the Bangladesh Rifles (BDR) said there were scattered exchanges of fire overnight and Thursday morning but said there were no new reports of casualties.
However fresh troops had arrived from nearby districts, and from the capital Dhaka, 240 km (150 miles) away, and the neighbouring outpost had been put on high alert, he said.
Long-running dispute
The new clashes come amidst a tense stand-off at another border outpost in the north-east Indian state of Meghalaya.
India says Bangladeshi troops have seized the border village of Pyrdiwah, near the town of Dauki, and are holding at least 20 soldiers hostage.
Areas of that particular stretch of border have been in dispute since Bangladesh independence in 1971. The ownership of villages on both sides of the border is claimed by both countries.
Pyrdiwah has been in the possession of India since 1971, although it is claimed by Bangladesh.
Commanders of the border forces of each side are now holding a meeting to resolve the crisis but correspondents say no progress has been made.
Skirmishes are relatively common between the two countries, which share a 4,000 km (2,480 mile) border, but the latest violence is especially serious.