Security forces are thought to have cornered the group of about 200 people in a district of Yangon since Monday.
The UN Human Rights Office said the group had been protesting peacefully and should be allowed to leave.
Mass protests have been seen across Myanmar since the military seized power on 1 February.
More than 54 people have been killed by security forces in demonstrations so far.
According to the UN, the group were blocked from leaving a four-street area in the Sanchaung area of the city on Monday.
Police have been raiding houses in the area looking for people who are from outside the district. Residents and a local news service claimed on Facebook that at least 20 people have been arrested in the raids.
Explosions have been heard from the area, believed to be the sound of stun grenades used by the military.
UN chief Antonio Guterres was calling for “maximum restraint” and the “safe release of all without violence or arrests”, his spokesman Stephane Dujarric said.
“Many of those trapped are women who were peacefully marching in commemoration of International Women’s Day,” he said.
The British Embassy also called for the protesters to be freed.
In Yangon, huge numbers of people gathered on the streets, defying a curfew, in an attempt to distract security forces. They were heard chanting “Free the students in Sanchaung”.
Security forces fired guns and used stun grenades in an attempt to disperse them, Reuters news agency reports.
It’s thought that three people died in demonstrations across the country on Monday. Protesters have been taking to the streets for the past month calling for an end to military rule and the release of the country’s elected government leaders – including Aung San Suu Kyi – who were overthrown and detained in the coup. (Courtesy BBC)