Dumisani Nkwamba, spokesperson for Public Service and Administration Minister Richard Baloyi, said the government will be left with no choice but to implement the deal if unions do not accept it in less than 21 days.
On Thursday, government said it had unilaterally signed the draft wage deal, warning that if unions failed to sign the deal, it will be "implementable".
Nkwamba said the government is determined to implement the offer, reiterating that if government were to meet unions' demands it would put a burden on the fiscus.
The government is offering a 7% wage increase and R700 a month housing allowance, while unions insist on an 8.6% wage increase and R1 000 housing allowance.
But unions insist that government should revise the wage package.
Mugwena Maluleke, speaking on behalf of unions affiliated to trade federation Cosatu , said they would embark on an industrial action until the government has a revised offer.
Maluleke said unions would respond to government's unilateral decision to sign the wage deal by intensifying the strike.
The unions and government had no scheduled formal negotiations, Maluleke said.
In its third day, Maluleke said the industrial action had moved into an even
higher gear.
About one million government workers embarked on industrial action from Wednesday, after rejecting government's latest wage offer.
The indefinite strike, which commenced on Wednesday, has disrupted public services in schools, hospitals, border posts and prisons.
Nkwamba said Baloyi intended to engage union leaders in a bid to resolve the
wage dispute.
The minister, Nkwamba said, is meeting union leaders to make them understand government's position and to urge them to reconsider the current offer.
But unions have hardened their stance even further.
The South African Medical Association (Sama) said it supported the demand of an
8.6% increase in wages.
"We support the specific demand for a salary increase but as an essential service we are unable to participate in the strike action," Sama said.
National Education, Health And Allied Workers Union (Nehawu), a Cosatu affiliate, said the war that unions were fighting was "the war that has been forced upon us".
Nehawu said the unions' initial demands were an 11.5% wage increase and a R2 000
monthly housing allowance.
"We reviewed our demands during the first round of negotiations after having received a presentation from the employer about the lack of financial resources to meet our initial demands," Nehawu said.
"Our current demands of 8.6% and R1 000 housing allowance signified a huge drop and compromise on the side of the workers and it will be disingenuous for anyone to say otherwise."