Chad has accused Sudan of backing a rebel group's push into the country.
The allegation comes just two days after the two countries signed an agreement in the Qatari capital to end hostilities against each other and normalise relations.
A deal brokered by Qatar and Libya was signed in Doha on Sunday to produce a lasting settlement to a six-year-old uprising in the western Sudanese region of Darfur, where Chad's rebels have rear bases.
Chadian officials say hundreds of armed men and vehicles crossed the border from Sudan and are now stationed near the key towns of Abeche and Goz Beida in the east of the country.
"For the moment, there is no major interruption to humanitarian activities, essential services remain active," said Katy Thiam, a spokeswoman for the UN Office for the Co-ordination of Humanitarian Affairs, from Abeche, the main town in east Chad.
The Union of Resistance Forces (UFR), the Sudan-backed rebels, confirmed on Wednesday that their fighters had clashed with Chad's government forces in the southeastern Salamat region.
He said they were moving towards the capital Ndjamena.
"UFR forces continue to progress towards total control of Chad's main towns," Ali Ordjo Hemchi, a spokesman for the rebels, said in a statement.
Another rebel official told the AFP news agency by telephone: "We are doing everything in our power to reach Ndjamena. Our final objective is Ndjamena."
In the statement, Hemchi said the rebels captured 12 army vehicles and destroyed nine others in Tuesday's clash between the towns of Tissi and Haraz-Mangue, claiming the government troops fled.
He gave no details of casualties on either side.
The African Union on Wednesday expressed "deep concern" over the offensive, which came as a serious blow to recent efforts to normalise the long-strained ties between the two neighbours.
"The chairperson of the Commission of the African Union, Jean Ping, follows with deep concern recent developments in eastern Chad, especially reports of incursions of armed rebels in this region," a statement said.
Chad's government announced on Tuesday that the rebels had launched an assault backed by Sudan, using several hundred vehicles.
It accused Sudan of reneging on a weekend peace agreement. Khartoum has denied any part in the offensive.
Against this backdrop of renewed fighting, Bernard Kouchner, the French foreign minister, met his Chadian counterpart, Moussa Faki Mahamat, on Wednesday, the foreign ministry announced, giving no details of the substance of the talks.
France has 1,100 soldiers serving in its former colony under a bilateral accord and 800 of its troops are serving in a UN-led force that last month took over a European mission to protect refugees in camps.
In February last year, rebels battled their way to the outskirts of Chad's capital Ndjamena in a bid to overthrow President Idiss Deby Itno before being beaten back with logistical help from some French forces.
That fighting delayed deployment of a European military force, Eufor, sent in to protect humanitarian staff and some 450,000 refugees from Darfur and displaced civilians in both Chad and the neighbouring Central African Republic.
A UN mission in Chad (Minurcat), which took over from Eufor in March, says it has stepped up military patrols around Goz Beida and also restricted the movements of humanitarian personnel.