The Philippines denied the claim, saying Duterte had simply pledged to respect whatever the outcome of Indonesia's judicial process.
Duterte - who is waging a bloody crackdown on drugs in his country that has claimed nearly 3,000 lives in just two months - apparently gave Widodo the green light during a bilateral meeting last week in Jakarta.
Widodo said he and Duterte were discussing the woman's case when the controversial Filipino president made the remarks.
"President Duterte then said please, go ahead if you want to execute."
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Mary Jane Veloso was arrested in Indonesia in 2009 carrying a suitcase lined with 2.6 kilograms of heroin and was sentenced to death.
The mother of two was moments away from facing the firing squad last year when she was granted a temporary reprieve, after a woman suspected of recruiting her was arrested in the Philippines.
Legal proceedings against the alleged recruiter are ongoing in the Philippines. The Indonesian government, which has some of the world's toughest anti-narcotics laws, maintains Veloso's death sentence for drugs trafficking still stands.
Veloso's execution had been "indefinitely deferred", Yasay added.
Her case attracted huge attention in the Philippines and Indonesia, with rallies of support and world boxing superstar Manny Pacquiao pleading for her life to be spared.
Her supporters claim she went to Indonesia for a job as a maid and was duped by an international drug syndicate into carrying heroin.
Widodo is waging his own war against drugs and has ordered the execution of eighteen convicted smugglers - including 15 foreigners - since taking office in October 2014.
The country's anti-drugs chief this month voiced support for implementing a bloody crackdown on traffickers like that in the Philippines, saying he believed such a campaign would safeguard "our beloved country".
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