"This is our biggest operation in several years; it started Sunday, is continuing today, and we don't know how long it will last," said Salah Barqueen, a member of the Asmara-based Eastern Front central committee.
"We have captured 20 prisoners, including one officer, and are now 30 kilometers (18.5 miles) from Tokar," he told AFP. Tokar is about 120 kilometers (75 miles) south of Port Sudan.
Sudanese government officials in Khartoum could not immediately be reached to comment on the claims and rebels did not provide any details on the number of fighters involved.
An Asmara-based JEM leader confirmed his group's forces were involved in the operation and stressed that his movement was not solely a Darfur-based organisation but had branches across the country.
"We want a comprehensive solution to the problems of Sudan," Abdelaziz Osher told AFP.
"As long as the government offers a peacemeal solution, even is there an agreement on Darfur, we will continue the struggle somewhere else; in the East, in Khartoum, in Kordofan," he said.
Rebels both in Darfur and in the East claim the government has marginalized their regions.
In April, the newly-formed Eastern Front vowed to carry on the fight against Khartoum at a meeting with others in rebel-held territory in eastern Sudan near the Eritrean border.
"It's not the first time we have had joint missions, because we are fighting the same target: the Khartoum government," Barqueen said, adding that the offensive would continue "depend(ing) on the results we get."
Asked if the Eastern Front's aim was to reach and take Port Sudan, the country's main port, Barqueen demurred, answering only: "Our objectives are many."
Earlier this month, the Eastern Front said it had captured three Sudanese legislators in Eastern Sudan in May and were holding them as "prisoners of war" in what appeared to be their first overt military act against the government since the April meeting.
The Eastern Front was formed in February by the Beja Congress, its main component, and the Free Lions, with the support of assorted other factions.
The Beja Congress first took up arms against Khartoum in 1994 and now controls the region north of the town of Kassala near the Eritrean border. Since 1994, fighting has been sporadic in the region.
There had been signs that the January 9 signing of a peace deal between Khartoum and southern Sudan rebels could lead to an end to the struggle in the east but those hopes all but collapsed later that month when government forces used deadly force to disperse a riot of Beja Congress supporters in Port Sudan.
Between 14 and 36 people were killed in the incident after which Khartoum pledged to hold talks with the rebels.
That offer has been met with widespread skepticism and the talks have yet to materialize with the eastern rebel groups vowing to continue their struggle and defend themselves from the government army and its proxy militia.
Barqueen said the current offensive was a direct result of Khartoum's failure to abide by its promises.
"The Khartoum regime is not genuine, we have had no indication of peace talks to come, so we have started this military plan," he said.