The Polish Border Guard (SG) tweeted it had detained a group of Iraqi migrants half a kilometre from the border.
One of the men has died despite attempts to resuscitate him by SG officers and paramedics, the agency said.
The death is suspected to have been caused by a heart attack, state news agency PAP reported.
Another member of the group has been taken to hospital after testing positive for COVID-19, the Polish Border Guard also said.
Anna Michalska, the spokeswoman for the agency, told the PAP news agency that coronavirus tests had been administered to all those detained.
The incident took place in Poland's northeastern province of Podlaskie, near the village of Nowy Dwór, which houses one of the Border Guard’s outposts, PAP reported.
Last Sunday, the Polish border service reported the discovery of the bodies of three migrants on the Polish side of the frontier. The most likely cause of those deaths was hypothermia, the Polish state news agency reported, citing prosecutors.
Meanwhile, the Polish Border Guard reported this week that a woman had been found dead on the Belarusian side.
Prime Minister Mateusz Morawiecki said on Monday that the ongoing migrant crisis on Poland's eastern border with Belarus had been created to destabilise Poland and the rest of the European Union.
Poland and fellow EU members Lithuania, Latvia and Estonia have accused Belarus' strongman leader Alexander Lukashenko of organising a wave of illegal migrants seeking to enter the bloc as part of what officials have called a "hybrid war."
At the start of this month, Poland's President Andrzej Duda declared a state of emergency in parts of two regions along the country's eastern border with Belarus in a bid to stem the flow of migrants from countries such as Iraq and Afghanistan.
The 30-day state of emergency gives authorities broader powers to monitor and control the movement of people.