Three senior prison officials, including the director of the Altiplano jail, have been dismissed.
It has emerged that Guzman fled despite wearing a monitoring bracelet and being subject to 24-hour surveillance.
"There will be no rest for this criminal," said Mr Osorio Chong.
On Monday, he visited the prison and the nearby area for the first time since Guzman`s escape.
The interior minister arrived by helicopter at the Altiplano compound, about 90km (55 miles) west of Mexico City, along with a group of congressmen and National Security Commissioner Monte Alejandro Rubrido.
Mr Osorio Chong is in charge of the operation in the absence of President Enrique Pena Nieto, who is on a planned visit to France for that country`s Bastille Day celebrations on Tuesday.
Laundry basket escape
Security has been reinforced across Mexico. Flights from nearby Toluca airport were suspended and checkpoints have been set up.
Guatemala also increased checks along its northern border with Mexico in response to the news that Guzman had escaped. It was there he was captured in 1993.
Guzman was sent to a top-security jail in Mexico, Puente Grande, but broke out in 2001, reportedly hidden in a laundry cart after bribing officials.
In the following years, he expanded the reach of his organisation, says the BBC`s Juan Paullier in Mexico City.
Analysts are warning that the same could happen again, if Guzman`s escape triggers conflicts with other cartels keen to protect their patches.
Dozens of prison guards are being questioned at the Altiplano prison to find out how he could escape this time.
Video footage showed Guzman entering the shower area in his cell at 20:52 local time on Saturday (01:52 GMT Sunday).
The tunnel linked the shower area to a house built over the past year across maize fields near the prison.
The tunnel, which measures 1.7m by 70cm, would have allowed Guzman, who is known as El Chapo or Shorty, to comfortably walk upright.
It had ventilation and lights and guards also found a motorcycle which they believe was used to transport earth removed as the tunnel was dug.
Inside the building near the prison, officials found a bed and a kitchen, suggesting those who dug the tunnel could have spent days at a time there.
A calculation based on the height, width and extension of the tunnel estimates that the earth removed would have filled 379 lorries, said Reforma newspaper.
President Pena Nieto called Guzman`s escape "an affront to the state" and ordered a full investigation.
Guzman`s personal fortune is estimated at $1bn (£640m).
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