Labour's John McDonnell has suggested voters may be ready to pay more council tax, as polling stations open for local elections in England.
The shadow chancellor said the message from the doorstep was that voters were concerned about the fabric of their communities and many were saying they would "rather pay a bit more to get a decent service".
Labour is widely expected to make gains in Thursday's ballots, which put Theresa May to her biggest electoral test since she lost her Government majority in last year's General Election.
London is the major battleground after recent polls pointed towards a substantial swing to Labour.
Jeremy Corbyn's party will be hoping to win overall control of Tower Hamlets, Barnet and Hillingdon and make inroads into the flagship Conservative councils of Wandsworth and Westminster.
Upsets are also possible around the country, with Labour hopeful of taking power in town halls from Kirklees and Calderdale in Yorkshire and Trafford in Greater Manchester to Plymouth and Swindon in the south.
However, Mr McDonnell sought to damp down expectations, telling The Guardian: "It will be a good night, but we're being set up by the Tories and the media, so if there's a single Tory councillor left in this country we'll have failed."
Conservative chairman Brandon Lewis urged the party's voters to turn out, warning that many results may be tight.
"We have to fight for every single seat," Mr Lewis told The Sun. "Getting out the vote is all-important, there are councils which could be decided on hundreds of votes in various wards."
Writing in the Daily Mirror, Mr Corbyn blamed Conservative policies for increased death rates in the early part of this year.
"Tory austerity has almost certainly increased the death rate," said the Labour leader, urging voters to "strike a blow against these deadly policies".
Mr McDonnell said: "What people are saying is, the fabric of our community isn't what it was and it needs addressing.
"The council tax issue is interesting. We're getting on the doorstep, 'I'd rather pay a bit more to get a decent service'."
The elections will give voters a chance to pass their verdict on the Prime Minister's handling of Brexit, and Liberal Democrats are hoping their promise of a second referendum will help them regain Remain-backing Kingston and Richmond in south London.
Elsewhere, Conservatives may be boosted by Ukip's decline in the polls, with targets like Basildon or Great Yarmouth potentially up for grabs if the Eurosceptic party performs poorly.
Across England, more than 4,000 seats are being contested in around 150 councils - including all 32 London boroughs, as well as every ward in Birmingham, Manchester, Leeds and Newcastle.
Mayoral elections are taking place in Hackney, Lewisham, Newham, Tower Hamlets, Watford and the Sheffield City region, but there are no local authority polls in Scotland, Wales or Northern Ireland.
Recent national opinion polls have seen Conservatives and Labour running neck-and-neck on about 40% each.
But a survey last week for Queen Mary University gave Mr Corbyn's party an overwhelming 51%-29% lead over the Tories in London, enough to deliver the best result in the capital for any party in more than 40 years.
Even if these figures are repeated on Thursday, the prize of Westminster - Tory-held since the council was created in 1964 - may remain beyond Labour's reach.
But Wandsworth, which has been in Conservative hands since 1978 and was once regarded as Margaret Thatcher's favourite council, may be more vulnerable to a Labour surge.
Another long-standing Tory citadel, Kensington and Chelsea, is a wild card in 2018, due to local anger over the council's handling of the Grenfell Tower fire tragedy.
New party Advance Together has 19 candidates fighting on a platform of "working for the interests of all residents, not for a political party".
Other areas where a swing to Labour could deliver overall control to Mr Corbyn's party include Amber Valley in Derbyshire, Carlisle, Dudley, Walsall, Worcester and North-East Lincolnshire.
Councils to watch for possible Conservative gains include Portsmouth, Basildon, Thurrock, Peterborough and Colchester, all of which could see the party win overall control on a relatively small swing.
The British National Party may end with no representation on any council in the UK, as its sole current councillor is not defending his seat in Pendle, Lancashire.
Most councils are counting votes overnight but others will declare results during the day on Friday.
Polls close at 10pm and the first results could begin to arrive at around midnight and are expected to be in full flow by 2am.
Controversial plans forcing voters to prove their identity before casting their ballot will be trialled at some polling stations in Bromley, Gosport, Swindon, Watford and Woking.
Ministers say the scheme will help combat electoral fraud but critics argue it will suppress turnout.