The Sudan People's Liberation Movement-North (SPLM-N) said they were just a few kilometres from El Kurmuk, the major town in southeastern Blue Nile state.
"It's very close and we are controlling the Kurmuk airstrip," which is west of the town, rebel spokesman Arnu Ngutulu Lodi told AFP.
"This is a very serious one," he said of the battle.
The army denied there was fighting on the outskirts of Kurmuk but said its forces had withdrawn from an area southwest of the town which, two days earlier, it announced it had "liberated."
The United States said on Friday it was "deeply concerned" by reports of clashes between Sudan and South Sudan in disputed border areas between Blue Nile and South Sudan's Upper Nile state.
Khartoum accuses South Sudan of backing rebels in South Kordofan and Blue Nile, but the South denies such support.