Even if agreement is reached on the Cabinet, the ideological differences between the ANC and DA could hinder policymaking down the road, analysts say.
A DA spokesperson said the party would communicate officially when negotiations were finalised.
Once unthinkable, the accord allowed President Cyril Ramaphosa to win a second term in office. He was re-elected by lawmakers with 283 votes.
The DA, the largest opposition party, has said it won't join a formation with the EFF or former President Jacob Zuma's uMkhonto we Sizwe (MK) party, which came in a surprising third in last month's vote.
The ANC remains the biggest party, but the lack of an outright majority means it must now share power, and President Cyril Ramaphosa says its leaders decided that a broad collaboration was the best way forward.
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 result means that the ANC must now share power, likely with a major political rival, in order to keep it - an unprecedented prospect in South Africa's post-apartheid history.
The ANC has lost its outright majority in parliament for the first time since it came to power after apartheid. In the previous election in 2019, the party won 230 seats.
For the first time in South Africa's democratic era, the African National Congress (ANC) will have to seek one or more coalition partners to govern with after it fell well short of a majority in last week's national election.
Mwaura said the elections will coincide with those of other wards in Nairobi, Narok, West Pokot, Homa Bay and Busia counties which were cancelled due to flash floods.
The minority UDA side has split into two factions that are both fighting for leadership roles.
Joho wants to succeed opposition chief Raila Odinga in leading the Orange Democratic Movement.
Chris Pappas has risen above the identity politics of KwaZulu-Natal to become mayor of the uMngeni Municipality and a candidate to perhaps run the entire province.