The men were detained by police after deciding not to board their plane.
They are reported to have refused to go home despite the intervention of the Eritrean ambassador in Botswana.
There have been a number of mass defections by Eritrean athletes in recent years.
One of the most noticeable was in 2013, when Uganda granted asylum to 15 Eritrean players and their team doctor after they absconded at the end of a football tournament.
A senior official at the Botswana Football Association says it has yet to establish why the players refused to leave with their team-mates.
However a report in the Botswana newspaper The Voice said that the players were distressed about the prospect of joining the Eritrean army and wanted to seek asylum despite being persuaded not to by the ambassador.
The newspaper reported Botswana Football Association vice president Basadi Akoonyatse as saying that the Eritrean players were found missing from their hotel rooms early on Wednesday morning by team officials.
"A report was made to the police who spotted them loitering around Francistown, alleging that they were in search of a Red Cross centre where they could seek assistance to be accorded asylum in Botswana," she said.
Eritrea is notorious for human rights abuses, with torture and slavery both reported by the United Nations Commission of Inquiry report to be commonplace.
The country has also been accused by the UN of forcing its citizens to indefinite national service and of killing people trying to escape abroad.
The Eritrean government dismissed the UN's findings as "totally unfounded and devoid of all merit".
Nation of six million on Red Sea - one of Africa's poorest countries
One-party state - no functioning constitution or independent media
Former Italian colony, later formed loose federation with Ethiopia
1962 - Ethiopian Emperor Haile Selassie dissolved Eritrean parliament, seized Eritrea
Eritrean separatists - the Eritrean People's Liberation Front - fought guerrilla war until 1991, when they captured capital Asmara
Eritrea voted for independence in 1993
May 1998 border dispute with Ethiopia led to two-year war costing 100,000 lives
Still no peace settlement - thousands of troops face each other along 1,000km (620-mile) border
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