In a letter delivered to Online International News Network, a Pakistan-based news agency, the kidnappers threatened to kill Solecki within 72 hours unless Pakistani authorities released 141 women prisoners.
The commander of a government paramilitary unit in Quetta denied females were being held.
"Who are these 141 women? Who knows who they are? This is just attention-seeking. This is just terrorism," Major General Salim Nawaz told the Aaj television network.
The letter also said the group, which has identified itself as the Baluchistan Liberation United Front, would be issuing another list of hundreds of male prisoners who they also want freed.
"We will issue this list very soon. If our demands are not met, we will murder John Solecki," the letter said.
Farhan Haq, a UN associate spokesman, said: "We have seen the video and based on what we can see, it appears to be the person in the picture is John Solecki.
"We have seen the video ... it appears the person in the picture is John Solecki"
Farhan Hag, UN spokesman
"Our team in Islamabad continues to examine the information. We have nothing further to add at this time beyond reiterating our appeal that John be released."
Maki Shinohara, a spokeswoman for the UN, said it had not been able to establish contact with the kidnappers. She said officials were trying to work out how to respond to the threat and the demands.
Incidents of abduction have increased in Pakistan over the past year, especially in the northwest in areas on the border with Afghanistan, where several foreigners have been taken by pro-Taliban fighters.
A number of nationalist groups are fighting for autonomy in Baluchistan province, but such groups are not known for targeting foreigners.
The UNHCR has been working for years in Quetta with refugees from Afghanistan.