The strike was planned to take place next Tuesday, October 14, as part of a co-ordinated demand across the country for a ‘fair pay rise for Britain’.
A march and rally was planned for Sheffield.
Rod Padley, branch secretary of the Sheffield Unison branch, said: “It might have changed but the offer was 2.2 per cent over two years but bottom loaded so those at the bottom would get considerably more.
“There have been negotiations, that may have changed.
“If they manage to secure a new deal for our members then we will be very happy - nobody wants to strike.”
Unison, GMB and Unite said they had suspended the strike to consult on the new negotiations from the Local Government Association.
But a strike by staff at Sheffield’s five household waste recycling centres is expected to go ahead on Friday in a separate dispute.
And health workers are due to walk out in Sheffield on Monday, and civil service workers in the PCS union on Wednesday, in similar protests over pay.