'We are campaigning to remove the barriers that prevent people wanting to become councillors by enabling them to combine home life with councillor duties,' said LGIU communications director Laraine Miles. 'One of the ways that this could be achieved is through a Public Service Act.'
The body is concerned that too many people are deterred from standing by the difficulties of balancing councillor duties with full-time jobs.
A large proportion of UK councillors are retired.
Miles added: 'We want better allowances and support for councillors so more people with an interest in community affairs become involved in local government and receive adequate recompense for it. For instance, they should be entitled to contribute to the local government pension scheme.'
Connect is charged with helping the body boost relationships with MPs and helping it set up a House of Commons all-party local government group, similar to the existing all-party charities group, which is serviced by the National Council for Voluntary Organisations.
The LGIU acts as a policy and research think-tank for local authorities and provides member councils with advice and training.
In addition to Westminster lobbying, Connect has been briefed to bolster the body's membership by promoting the benefits of its expertise to council leaders and chief executives.
Connect won a five-way pitch for the contract, which is led by account director Andy Sawford and supported by consultants Sarah King and Tessa Rodewaldt.