The hotlines handle enquiries from vulnerable adults, children and homeless people, as well as some reports of anti-social behaviour, and were entrusted to South Yorkshire Police seven years ago to become part of the 101 non-emergency number at an annual cost of Â£252,000.
However, the council believes bringing the services back in-house will 'improve call performance' while making savings of almost Â£50,000 per year.
Police have been 'unable to consistently achieve' a target of answering 90 per cent of calls within 30 seconds, a report to the deputy leader and cabinet member for finance, Coun Olivia Blake, said.
The authority's own out-of-hours housing repairs service, based at Manor Lane, 'overachieves' by answering 96 per cent of calls against a benchmark of 85 per cent.
"Conversely, for the period February 2016 to February 2017, SYP call performance dipped as low as 23.1 per cent for its homeless service enquiries," said the report.
"These vulnerable callers were abandoning between 26 and 35 seconds, and then redialling as they are unable to get through."
Officers emphasised that the change would allow police to deal with 'immediate life or death anti-social behaviour calls, rather than being bogged down with non-core work'.
The 101 number went live in Sheffield in 2005 and has 'proved a lifeline for people who may not have previously reported issues to the council or police', the report said.
Sheffield Council was one of the only areas that kept up a partnership with the police after the initial Home Office funding was withdrawn. The three extra services were transferred in 2010.
No formal contract is presently in place, which allows the arrangement to be terminated at any time, with payments made to the police force on a monthly basis.
"In addition, despite the 101 service being available to all South Yorkshire residents - in line with the policing model - Rotherham, Barnsley and Doncaster Councils have never contributed to the funding of this service."
The Manor Lane call centre has 'capacity to deliver more work', the report said. "Insourcing is the current financially viable option."
South Yorkshire Police has been approached for comment.