The 82-year-old revealed on Monday that he had taken the decision to leave Spain to help his son, the current King Felipe VI, "exercise his responsibilities".
The letter, published on the royal palace's website, did not mention where the former king would go, nor when exactly he would leave the country.
But on Tuesday, daily newspaper ABC reported that he left Spain on Sunday and flew to the Dominican Republic via Portugal.
The La Vanguardia and El Mundo dailies similiarly said he planned to stay with friends in the Caribbean country, but online newspaper El Confidencial said he could be in Portugal, where he spent part of his youth, or in France or Italy.
The ex-king's lawyer, Javier Sanchez-Junco, issued a statement Monday saying his client was not trying to escape justice by going into exile and would remain available to prosecutors.
The former head of state has been under a cloud since various media reported that he allegedly received funds from Saudi Arabia and probes are now under way in both Switzerland and Spain.
Spain's Supreme Court announced in June an investigation to determine the legal responsibility of the ex-monarch -- but only for acts committed after his abdication in 2014, because of the immunity he holds.
The suspicions centre on $100 million (85 million euros) allegedly paid secretly into a Swiss bank account in 2008.
After a series of media revelations, Socialist Prime Minister Pedro Sanchez spoke out in July saying "the entire Spanish population are receiving disturbing information which is troubling for us all, including me".
Juan Carlos ascended the throne in 1975 on the death of the fascist dictator Francisco Franco and ruled for 38 years before abdicating in favour of his son Felipe VI in June 2014.
He was a popular figure for decades, playing a key role in the democratic transition from the Franco dictatorship which ruled Spain from 1939-1975.
If you were surprised Epstein committed suicide, imagine how surprised he was!