The country's main trade-union confederation, Cosatu, which represents 1.7m workers, hints that it may join in.
The SACP and Cosatu believe the government's market-friendly policies and tight spending limits are not addressing these social needs.
Because of Mr Zuma's closeness to Cosatu and the Communists, some predict that he may preside over a radical shift to the left.
"We are just too tired of getting blackmailed all the time, " Zwelinzima Vavi, the General Secretary of COSATU, told the BBC's World Business Report.
The above-inflation pay increases that Cosatu has claimed over the past five years have swelled government expenditure and crowded out spending on worthier causes.
The SACP and the trade union federation, Cosatu, have been among Mr Zuma's key supporters, and have also been strident critics of President Mbeki's economic policy.
In the past five years the Congress of South African Trade Unions (Cosatu) has been criticised by many of its members for not fighting their corner hard enough.
South Africa's trade union confederation, Cosatu, also expressed concern that Mr Tutu "has allowed his passionate personal hatred of the ANC and its President Jacob Zuma to cloud his judgement".
"When a very similar incident occurred in India recently, there was a massive outbreak of protest and mass demonstrations in the streets, " Cosatu spokesman Patrick Craven said, AFP news agency reports.
The Congress of South African Trade Unions (Cosatu) demanded an excessive 8.6% pay rise, more than double the rate of inflation, at a time when the country's recovery from last year's recession is fragile.
"Senior political leaders in the province in their majority are 'tenderpreneurs', " is the blunt assessment of Dan Sebabi, the head of Limpopo's branch of Cosatu, the coalition of trade unions that is in alliance with the ANC.
On Wednesday, the governing African National Congress (ANC) party and the Congress of South African Trade Unions (Cosatu) demanded an investigation into how the guests of the family flew from India to the Waterkloof Air Force Base near Pretoria.
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