Vietnamese Foreign Minister Pham Binh Minh made the request of his Malaysian counterpart Saifuddin Abdullah during their phone talks Wednesday, according to a Vietnamese news outlet.
Indonesia mounted a diplomatic campaign to free Aisyah, with the country's justice minister writing to Malaysia's attorney-general asking for her release.
Vietnam has repeatedly called on Malaysia for a fair trial of one of its citizens accused of the murder of Kim Jong Nam, the half-brother of the North Korean leader, the southeast Asian nation's foreign ministry said on Thursday.
Prosecutor Iskandar Ahmad did not explain the decision to continue the trial of Doan Thi Huong.
"Very obviously, there is discrimination". "More so in this case because two people were charged, but it was withdrawn against one but not against the other".
But Huong says she was duped by North Korean spies into approaching him, believing it was a harmless prank for a TV show.
Nam was assassinated with a liquid VX nerve agent in February 2017 at Kuala Lumpur airport.
He also sought a deferment of the trial, saying Huong has been unwell since Aisyah's release and is not in a position to testify.
When asked by the judge if she was unwell, Huong stood in the dock and said she suffered from tension and stress.
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Kim Jong-nam was the older half-brother of North Korea's authoritarian ruler, Kim Jong-un.
"I have no idea what is going on", a exhausted and pale-looking Huong, who was wrapped in a red headscarf and a coat, said through an interpreter.
The next hearing will take place on April 1. She could be seen shaking as she pleaded with Vietnam embassy officials.
He said he believes his daughter is innocent and told her to "stay calm and hope for the best outcome".
Huong's lawyers said they were deeply disappointed with the decision and the attorney general had a "moral obligation" to explain the reasoning given the global nature of the case.
Prosecutors have said the women were trained killers who knew they were holding poison.
Huong is now the only person on trial for the Cold War-style killing, and could face the death penalty if convicted. The other, Indonesian national Siti Aisyah, was freed Monday in a surprise ruling and returned to Indonesia hours later.
As of late Thursday night (local time), the Attorney-General's Chamber had not issued a statement about the decision in Doan's case.
It may be Seoul's reticence itself that should be viewed as inappropriate, given that two years ago, it denounced Kim's murder as an "explicit act of terror".
During the trial, some of the evidence against Ms Huong had seemed stronger - airport CCTV footage shown in court captured her approaching Mr Kim, placing her hands on his face and then running away.