The continent's first multi-sports gathering was hosted in Baku, Azerbaijan last year at a cost of several billion dollars but finding a second host has proved a major challenge.
"The acting president of the EOC is having talks with President Lukashenko as we speak to see if Minsk will host it. It is very likely but nothing has been signed yet," the source, who spoke on condition of anonymity, told Reuters.
Lukashenko is also the head of the country's Olympic committee.
Netherlands withdrew from hosting the second edition, scared off by the cost of the event, while second option Russia was ruled out following the massive state-backed doping scandal this year.
A rival European Sports Championships have been planned for 2018 in Berlin and Glasgow, with seven federations, including athletics and swimming, signing up for it.
Both these European federations refused to send their top athletes to Baku and are likely to repeat that stance for the 2019 gathering, a major setback for the event, with both sports among the most popular.
The arrest and brief imprisonment of suspended EOC chief Patrick Hickey during the Rio de Janeiro Olympics and subsequent charges of an alleged ticketing scam further hampered efforts to appoint a second host.
(Reporting by Karolos Grohmann; Editing by John O'Brien)
