Janusz Walus, a far-right immigrant from Poland who shot Hani dead in 1993, had been due to be released by Thursday after he was granted parole by the country's top court.
The decision sparked fierce protests from the ruling African National Congress (ANC) and its ally in the struggle against apartheid, the South African Communist Party (SACP).
"By agreement between his legal representatives and the state attorney, the matter of Mr Walus' parole will only be finalized after he has received the necessary medical clearance from medical team," justice ministry spokesman Chrispin Phiri said in a statement.
Walus is being held at the Kgosi Mampuru II Correctional Centre in Pretoria.
The Constitutional Court on November 21 granted Walus parole and gave the prison service 10 days in which to release him.
He was attacked by another inmate earlier this week while queueing for food.
Hani, a hugely popular figure and fierce opponent of White rule, was killed in his driveway just as negotiations to end apartheid were entering their final phase.
The SACP, which Hani used to head, has petitioned the court to go back on its ruling.
The murder almost plunged South Africa into a race war.