He also wants the investigation into the case sped up, reports the Bangkok Post.
The prime minister gave the order at a cabinet meeting after the RTP provided an update on the probe, said government spokesman Anucha Burapachaisri.
As for tracking down Vorayuth, Mr Anucha said Gen Prayut has continuously ordered the RTP to try and bring Vorayuth back to the country.
The RTP has also been instructed to expedite the investigation into state officials and other individuals associated with the case, Mr Anucha said.
Mr Anucha did not go into details about the updated investigation.
Vorayuth has been on the run since his Ferrari fatally hit a Thong Lor police officer on Sept 3, 2012.
Vorayuth was charged in August 2020 for cocaine abuse in addition to reckless driving causing death, by a panel of prosecutors chaired by Itthiporn Kaewthip, director-general of the Office of Criminal Prosecution.
A court warrant was then issued for his arrest under Section 58 of the earlier Narcotics Act of 1979, which prohibited the use of cocaine.
Offenders were liable to a jail term of between six months and three years and the statute of limitations was 10 years, which would have been Sept 3, 2022 for Vorayuth.
But when the new narcotics law took effect on Dec 9 last year, the Narcotics Act of 1979 was cancelled.
Under Section 162 of the new law, which also prohibits the use of cocaine, a violator is liable to a jail term of up to one year and the statute of limitations was reduced to five years.
However, a speeding charge against him was dropped after its one-year statute of limitations expired in 2013, followed by another charge of failing to help a crash victim, which expired in 2017.
The only remaining charge against him is reckless driving causing death, which carries a penalty of up to 10 years in jail and has a 15-year statute of limitations, which will expire in 2027.
In May, a panel determined the fate of Nate Naksuk, the former director-general of the Office of the Attorney-General (OAG), who dropped a charge of reckless driving causing death against Vorayuth after the speed at which he was driving was reduced to an estimated level of below 80kph. An initial report showed Vorayuth drove his Ferrari at 177 kilometres per hour at the time of the alleged accident.
The remaining charge was filed recently by public prosecutors.
Public Prosecutors Commission chairman Patchara Yutithamdamrong said a meeting found that Mr Nate had handled the case with gross negligence, severely damaging the OAG’s reputation and authority under Sections 85 and 87 of the Civil Service Act 2010.
The meeting agreed Mr Nate deserved the harshest penalty of being dismissed. But they later decided to let him keep his pension.