The authority said it had put forward new proposals which would see PFI contractor Amey spending more money to retain more trees.
The plans will now be looked at in the coming weeks by campaigners from Sheffield Tree Action Group (STAG), who have been fighting to save the street trees. The potential compromise came about after mediation talks between the two sides.
Councillor Lewis Dagnall, Sheffield City Council’s Cabinet Member for Environment and Streetscene, told the BBC: “We’ve put forward a proposal on behalf of Amey and the City Council which would see fewer trees replaced over a phased period.
“We are now going to go into a period of a few weeks to give STAG the chance to further scrutinise the proposals that we’ve put forward before we then have a wider conversation with the public about the next steps.”
Asked whether this meant fewer trees would be felled, he said: “We have got a compromise from Amey which means that Amey have offered to take on more risk and spend more to retain more trees upfront.”
The council and STAG have, for the first time, been trying to reach a compromise after years of disputes over the issue that included legal action being taken in the High Court.
Sheffield City Council has signed a £2.2bn contract with Amey to improve the city’s highways over 25 years, which included a target to fell 17,500 street trees.
During the first five years of the contract, clashes have resulted in arrests of protesters and the council taking members of the group to court for standing in safety zones during works.
The first mediated talks between the two sides took place last month, overseen by the Bishop of Sheffield, Dr Pete Wilcox.