Dozens of services across the region are not running, affecting tens of thousands of people.
Members have been offered a new pay offer by Stagecoach and will suspend the strike to vote on it.
The offer has only been made to the depots in Rawmarsh and Barnsley. Strike action will continue in Sheffield and elsewhere in South Yorkshire.
The new pay offer comes following talks between Unite and Stagecoach mediated by conciliation service ACAS.
Unite regional officer Phil Brown said: “We will be making no further comment until the vote is finalised.
“Negotiations on behalf of our Stagecoach members in Sheffield are still ongoing however and will continue tomorrow [January 11]. There will be no suspension of strike action in Sheffield until an offer is put forward that our members could find acceptable.”
Meanwhile, Stagecoach claimed the undisclosed pay offer has a “strong union recommendation to accept”. THey will reportedly pick up talks on January 11 to seek a pay deal for Ecclesfield and Holbrook depots.
Phil Medlicott, Managing Director of Stagecoach Yorkshire, said: “We’re pleased to have held positive talks with Unite today, supported by the arbitration service ACAS, and have jointly reached agreement on a new pay deal for our employees at Barnsley and Rawmarsh."The positive talks today, which have come about as a result of both Stagecoach and Unite showing flexibility and working together to agree a new deal, will mean that the planned strike action impacting services from Rawmarsh and Barnsley has been suspended, which is great news for our local communities.
"Stagecoach and Unite will return to talks tomorrow with ACAS to try and agree a new pay deal for Ecclesfield and Holbrook Depots."
It comes after Mr Medlicott previously critcised the strike action as “unecessary” and that it “benefits no one".
Bus drivers have been at loggerheads with Stagecoach since October 2020 after a pay rise was rejected by union members.
The firm initially offered a two per cent increase. However, Unite labelled this a “poverty pay rise”, claiming its drivers had cumulatively been paid below inflation for over a decade.
A second pay offer offering a nine per cent rise within six months was also rejected.
Members walked out for two separate weeks in December, before beginning indefinite strike action on January 2.
Previously, Mr Medlicott has criticised the union. In a statement on December 19, the chief executive labeled the union’s rejection of the second pay offer as “counter to the spirit of achieving a jointly-brokered solution”.
ACAS is a non-departmental Government body set up to improve working relations between organisations. They are often brought in to mediate talks when firms are at loggerheads over strikes.