Both the Aftonbladet and Expressen newspapers report that the pair was found on Thursday afternoon. The woman was dead, while the man was still alive when police arrived on the scene.
He was taken to hospital but later died from his injuries.
Witnesses reported seeing a man fleeing the area and police suspect the killings occurred during an attempted robbery.
No suspects have yet been arrested.
Sweden’s foreign ministry learned of the murders on Thursday evening via Sweden’s honourary consul in Trinidad and Tobago.
“What we’ve heard is more hearsay. From what I know we haven’t yet received any account from the police,” said foreign ministry spokesperson Kent Öberg to the TT news agency.
He had no information about where in Sweden the couple lived.
According to Aftonbladet, the attack occurred in the home in Tobago which the couple had purchased four years ago.
The paper also reports that the murdered woman was 62-years-old and the man was 73.