Liberia, which is itself struggling to end a deadly two-year-old outbreak of the Ebola virus, has never won an Olympic medal. It had planned to send around 20 people to the games, about half of them athletes.
If Liberia does indeed decide not to go to Rio, it would be the first country to decline to attend on the grounds of the Zika outbreak in Brazil.
The finance ministry letter, dated March 21 and addressed to Liberia's deputy sports minister, also refers to a request for the approval of $151,882 (106,816 pounds) to send Liberian athletes to the games.
"Given the current fiscal position of the Government ... we are (unable) to grant this request at this time," read the letter, obtained by Reuters and authenticated by a sports ministry official.
"It is important to note that Brazil is currently battling the Zika epidemic ... and given our recent experience with Ebola, it will be public health risk to send (athletes) to that part of the world," it continued.
Philipbert Brown, president of Liberia's Olympic committee, said he had not yet seen the letter.
"But it would be disappointing if we do not attend the Olympics," he said. "Our people have been training for four years, and you come to say that you would not sponsor them?"
The Zika virus is believed to be linked to an explosion in cases of microcephaly among infants in Brazil. The government says 1.5 million people may have been infected nationwide.
The United States-based Centers for Disease Control and Prevention has advised pregnant women against travel to the summer games but the World Health Organization said the outbreak should neither affect international travel nor bar a successful Olympics.
More than 11,300 people have died in Liberia, Guinea and Sierra Leone in the past two years in the world's worst Ebola epidemic.
Liberia on Friday reported a third flare-up of the disease since the outbreak was first declared over in May.
The epidemic has also crippled the Liberian economy.
(Reporting by Alphonso Toweh; Writing by Makini Brice; Editing by Joe Bavier and Andrew Roche)
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