Jung has directed District Magistrate (South West) Ankur Garg to probe the matter and sought an inquiry report within a week.
Delhi Home Minister Satyendra Jain, however, said the probe should be entrusted to Divisional Commissioner (west district) B S Jaglan as Tihar Jail falls under his jurisdiction
He said the state government was yet to issue a written inquiry order.
While one of them was later caught, the other one is still on the run and a mega manhunt has been launched to nab him.
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"Ankur Garg, IAS, District Magistrate (South West) will conduct the inquiry into the incidents of wall break, digging of tunnel and escape of prisoners from Central Jail," LG's office said in a statement.
According to LG's office, Garg's inquiry will focus on the factors and circumstances leading to the incident besides reviewing comprehensively the security set-up of Delhi Prisons.
"District Magistrate (South West) may co-opt officers from concerned departments viz - Delhi Prisons, Delhi Police, PWD etc. To assist him in the inquiry," the LG's office stated.
Meanwhile, blaming administration for the incident, Kiran Bedi, former Inspector General Prisons of Tihar Jail today twitted "the escape reported from Tihar Prison is indicative of weakening of certain administrative & oversight practices, when followed, cannot occur.