It was unclear whether the aid had been dispatched already, as Mr. Putin seemed to imply in his conversation with Mr. Barroso, or would be sent later, after the terms of its delivery had been agreed upon.

“Very intensive work is being carried out,” said Dmitri S. Peskov, the presidential spokesman, according to Russian news agencies. Once the details were worked out, the aid would be delivered immediately because of the “tragic humanitarian situation” in Luhansk and Donetsk.

There is little doubt that Russia has the troop strength to carry out an invasion of southeastern Ukraine. NATO has said in the past that Russia had amassed about 20,000 troops along the border. On Monday, however, a Ukrainian military spokesman, Andriy Lysenko, said the numbers had risen alarmingly, to around 45,000 troops, supported by 160 tanks, 1,360 armored vehicles, 390 artillery systems, 150 truck-mounted ground-to-ground rocket launchers, 192 fighter jets and 137 helicopters. Those figures could not be independently verified.

Ukraine, the United States and European nations have repeatedly warned Russia against mounting a stealth invasion under the guise of humanitarian aid, and have looked on with growing alarm as Russian officials have spoken in ever-stronger terms about the humanitarian plight of eastern Ukrainians.

“We see the Russians developing the narrative and the pretext for such an operation under the guise of a humanitarian operation, and we see a military buildup that could be used to conduct such illegal military operations in Ukraine,” the NATO secretary general, Anders Fogh Rasmussen, told Reuters in an interview Monday.

A Russian intervention would almost certainly provoke another round of economic sanctions by the West, further isolate the Kremlin diplomatically and strain already icy relations with Washington nearly to the breaking point. Apparently calculating that Moscow is unwilling to shoulder those costs, Ukraine has been pressing ahead with its military offensive despite the Russian military buildup.

The Ukrainian offensive continued on Monday, as its military shelled outlying districts of Donetsk, tightening a siege there and on Luhansk, cutting supply lines for both civilian and military goods. Adding to a sense of desperation, residents in Donetsk awoke to learn that a stray rocket had broken open a gate of a high-security prison, allowing more than 100 convicts to escape.