Police found a suicide note in the home in the Mijas district of Torrenueva where the bodies of the three family members were found on Wednesday, Spain's ABC newspaper reported on Friday.
"Our life has become suffering, and we can't take it anymore," the note read, according to the national daily, which went to the indicate the deaths were a murder–suicide.
Also on Friday, the UK's Daily Mail newspaper named the three family members as Philip Wood, 56, his 54-year-old wife Sheila Wood, and their daughter Sophie, who was 27.
In the Daily Mail article, Sheila's Wood's brother alleged that her husband had killed both her and their daughter before committing suicide.
He thought it was the "only way out", the brother was quoted as saying.
The brother said both financial and health problems had made life increasingly difficult for the Woods.
Philip Wood had lost his job in April 2012 while his wife was facing imminent kidney failure, the brother said.
The brother said the murder–suicide must have seemed like the only solution.
The Wood family had lived in Spain for 10 years, family in Ireland said.
According to Spain's ABC newspaper, police investigators were "clear from the start" that Wood had used the same weapon to kill himself as the one he used to murder his wife and daughter.
The paper also indicated that possible financial reasons were being looked at as a motive.
The daily said toxicology reports would if Sheila and Sophie had been sedated and sleeping at the time they were shot.
Both the Daily Mail and ABC newspapers stressed the Sophie Wood had brain damage, and not Down's syndrome as had been previously reported.