A DA spokesperson said the party would communicate officially when negotiations were finalised.
The SABC report came as the newly elected parliament was convening for the first time and lawmakers were in the process of being sworn in.
The electoral math has created a complex situation for the ANC, which will have 159 of the 400 seats in the new National Assembly.
The ANC's potential partners are diametrically opposed, ranging from the free-marketeer Democratic Alliance (DA) to uMkhonto we Sizwe (MK) and the Economic Freedom Fighters (EFF).
The party and country will now have to enter a period of coalition building unprecedented in the democratic era, with potential partners ranging from the pro-business Democratic Alliance to insurgent parties led by former ANC figures who fell out with the party.
Some political analysts say this could lead to party leader and South Africa's President Cyril Ramaphosa having to step down.
Zuma, who was forced to quit as president in 2018, has fallen out with the governing African National Congress (ANC) and has been campaigning for a new party called uMkhonto we Sizwe (MK) named after the ANC's formed armed wing.
The victims were among the tens of thousands of people who crowded into a football stadium in Durban on Saturday for the launch of the African National Congress's election manifesto.
The number of employed persons decreased by 22,000 to 16.7 million, says national statistics agency, StatsSA.