* Three-week strike in Africa's largest economy to continue
* Unions say need more time to consult with members
* Smaller union umbrella group splits over vote
(Recasts with rejection by union coalition)
By Peroshni Govender and Jon Herskovitz
JOHANNESBURG, Sept 1 (Reuters) - South Africa's public sector unions rejected a new pay offer from the government on Wednesday to end a three-week strike but asked for more time to consult their 1.3 million members in Africa's biggest economy.
The majority of unions under a coalition, which included South Africa's largest labour federation COSATU, voted against the new offer, although some smaller unions in the grouping were split in their vote.
"The unions met this afternoon and COSATU rejected the offer. The strikes will continue and we will continue to consult our members," COSATU Secretary General Zwelinzima Vavi told Reuters.
Members of the smaller Independent Labour Caucus (ILC) union umbrella group said there was a split in their vote, although the group would go with the coalition majority.
"Irrespective of what we say, the majority rejected it. But we must still go to our members and get their mandate," said Manie de Clercq, a spokesman for the Public Servants Association, the biggest union affiliated to the ILC.
The strike has ended the national euphoria over hosting the June-July football World Cup. It has shut schools, led to bodies piling up at state morgues and dampened investor sentiment.
COSATU said earlier on Wednesday it was suspending a one-day sympathy strike this week by all its member unions, including private sector workers, to give the state workers more time to consult.
Suspending the sympathy strike, which could have shut the vital mining sector, removes some of the pressure on President Jacob Zuma's ruling African National Congress to reach a deal fast.
The government may be hard-pressed to improve its offer significantly, as it has said its latest offer is well above inflation -- 3.7 percent in July -- and it cannot afford what it has already put on the table.
This could leave union leaders with a dilemma -- whether to try to persuade members to accept the offer or to press ahead with a long strike that leaves strikers without pay in pursuit of a deal that further strains state finances.
COSATU has more than 20 affiliated unions and claims nearly 2 million members.
The government offered state workers, including teachers, nurses, customs officials and office clerks, a pay rise of 7.5 percent and 800 rand ($108.5) a month for housing. The unions are demanding an 8.6 percent rise and 1,000 rand a month.
The COSATU leadership, which said its alliance with the ANC was on the verge of rupture, appears to have taken a more conciliatory line since Zuma heeded its call for high-level intervention and ordered ministers to negotiate a rapid end to the strike, analysts said.
"They are turning the taps off a little, but are doing it slowly," said Sakhela Buhlungu, an expert on organised labour at the University of Johannesburg.
COSATU has threatened a prolonged strike by all its members, including miners and factory workers, which would cripple the economy, if there is no resolution of the state workers' strike.
South Africa is the world's fourth largest gold producer and largest platinum producer. The country's biggest firms, such as Anglo Platinum, Impala Platinum and Harmony Gold Mining , have stockpiles of ore and would not be seriously affected by a one-day work stoppage.
In yet another dispute in the mid-year strike season, the National Union of Mineworkers said more than 8,000 workers at Northam Platinum would strike on Monday in a pay dispute.
Officials have said the government will probably be forced to make cuts elsewhere and borrow funds to pay for the state workers' eventual pay rise, making it more difficult to meet its goal of cutting a deficit of 6.7 percent of GDP. (Editing by Marius Bosch and Peter Graff)