Daily newspaper ABC reported Tuesday that Juan Carlos left Spain on Sunday and flew via Porto, in neighbouring Portugal, to the Dominican Republic. La Vanguardia also said he was in the Caribbean country, but only temporarily.
But El Confidencial newspaper said he could be in Portugal, where he spent part of his childhood, or in France or Italy, where he has family and friends.
The 82-year-old former king is credited with helping Spain peacefully restore democracy after the death of dictator Francisco Franco in 1975.
But marred by scandals in the later years of his reign, Juan Carlos in 2014 abdicated in favour of his son Felipe, losing the immunity from prosecution Spain's Constitution grants to the head of state.
In the wake of Juan Carlos's announcement, some people called for the monarchy to be abolished. The leftist political party Unidas Podemos, the junior member of Spain's coalition government, wants a public debate about creating a republic.
There is no reason at all to keep supporting a monarchy which doesn't possess minimum ethical standards, the party said in a statement late Monday.
But the Socialist party, which leads the government under Prime Minister Pedro Snchez, has showed no willingness to follow that path and has declared its support for Felipe. Even so, Snchez recently said he found the developments about Juan Carlos disturbing. In downtown Madrid, opinions were divided.
I think it is really bad that we let him go, said Sara Fernandez, a 38-year-old insurance worker. He should have to stay here, return the money and do prison time, like all Spaniards when you break the law. But janitor Mar Verdugo, 55, urged caution.
We are judging him without any evidence, so when there is a sentence we will see if he has done right or wrong, she said.
Juan Carlos's lawyer, Javier Snchez-Junco Mans, said in an emailed statement that the former king had asked him to make clear that even though he will be outside Spain he intends to be available to cooperate with the investigation.
A statement from Spain's general prosecutor's office in June said it was investigating whether Juan Carlos received millions of dollars in kickbacks from Saudi Arabia during the construction of a high-speed railway there by a Spanish consortium.
Since then, Spanish media outlets have published damaging testimony from a separate Swiss investigation into millions of euros (dollars) that were allegedly given to Juan Carlos by Saudi Arabia's late King Abdullah.
Juan Carlos allegedly then transferred a large amount to a former companion in what investigators are considering as a possible attempt to hide the money from authorities.
The companion, Corinna Larsen, is a Danish-German businesswoman long linked by Spanish media to the former king. Spanish prosecutors have asked her to provide testimony in the case in September in Madrid.