Munyaradzi Paul Mangwana, a chairman of the select committee in charge of the process, said there were disagreements over who should gather public opinions.
He told the state-run Herald newspaper the committee also had a funding crunch and still needed to buy recording equipment and pay for transport.
The constitution was key to the power-sharing deal signed last February.
An election is due to be held once the constitution is agreed on.
In June 2008, Zanu-PF leader Robert Mugabe won a disputed presidential election which was boycotted by the Movement for Democratic Change (MDC).
MDC leader Morgan Tsvangirai, now prime minister in the power-sharing government, has on several occasions accused Zanu-PF of holding up the consultation process.
Mr Mangwana said he could not give a timetable for when the operation might restart.
The Herald reported that all other programmes connected with the constitution would be suspended until next week, when the committee would meet to discuss financing.
The constitution is intended to replace the document drawn up when Zimbabwe gained independence from the UK in 1980.
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