LONDON (Reuters) - Prime Minister David Cameron said he was confident he could persuade voters to back deep public spending cuts next month that he hoped would reduce the countrys worst peacetime budget deficit. Unions are threatening coordinated strikes to protest about the cuts, but Cameron tried in a newspaper interview published on Saturday to reassure the public by promising to give something back once the deficit is under control and the economy recovers. Chancellor George Osborne will announce details of the cuts in parliament on October 20, in a statement that many believe will place huge strain on the coalition government. I think we can (win the argument) because the public do agree that the deficit is a problem and you have got to do the job properly, Cameron told the Daily Telegraph. There will be an ability over time once you have sorted out debts and deficits to give people back something from a growing economy ... share the proceeds of growth. A Reuters/Ipsos MORI poll earlier this month showed that three out of five voters agree that cuts are needed, but three-quarters think the savings should be made more slowly than planned by the government. Support for the Liberal Democrats, the junior coalition partner, has slumped since the election and some supporters are concerned about the anticipated depth of the spending cuts. The Labour Party has accused the coalition of risking the economic recovery by cutting services too quickly and too deeply. Most government departments will have to make cuts of around 25 percent in real terms over the next five years as ministers try to reduce a deficit that reached 11 percent of GDP, or 154.7 billion pounds, in the last financial year. In the Telegraph interview, Cameron gave few new clues about where the cuts would be made, though he confirmed the government was reviewing universal benefits such as child payments. You do have to look at whether we can afford to pay all the benefits and whether they are fair, he was quoted as saying.