The footballer, who spoke on condition of anonymity, told RFI that 17 footballers and the team doctor are hiding in Kampala.
"Everybody has to stay in secure places because the Eritrean government is searching for us," he said. "The Eritrean embassay in Uganda are trying to find us."
The team member said the group could make an asylum claim on Thursday.
He said he had decided to flee Eritrea because of its many problems, citing the government of President Isaias Afeworki as "the main problem" and accusing it of failing to respect human rights and having a compulsory and extensive programme of military training.
The Red Sea Boys were until now participating in the Cecafa Cup, a tournament organised by the Council for East and Central Africa Football Association.
Eritrean footballers have often defected in the past, leading to them being required to pay a "surety" before traveling abroad - and suffer the consequences if they failed to return to Asmara.
Four players of Red Sea FC sought asylum in Kenya after a CAF Champions' League match in Nairobi in 2006. Up to 12 members of the national team stayed behind after the Cecafa Cup in Tanzania in 2007.
Another six players remained in Angola after an Africa Cup of Nations match in 2008.
Eritrea Red Sea Boys played their last game with Rwanda at Lugogo Stadium in Kampala, losing 2-0.