By Kylie MacLellan

LONDON (Reuters) - Britain's influence in the European Union will be stronger if it votes to remain in the bloc in a June 23 referendum, Prime Minister David Cameron said on Sunday with the latest polls showing Britons almost evenly split over whether to stay or go.

One poll published late on Saturday gave a two-point lead to supporters of "Remain" and the other showed those in favour of Brexit were one point ahead.

In an interview on BBC television, Cameron - whose "In" campaign has been branded as scaremongering by pro-Brexit supporters for warning of the risks of quitting the 28-nation EU - highlighted the advantages of staying.

"If we wake up on June 24 and we are in, Britain's authority within the EU will be stronger," he said.

"No other country has done what we have done, hold a renegotiation (for improved EU membership terms), hold a referendum and then people will know that the British agenda ... is going straight to the top."

The referendum will be a decision with far-reaching implications for politics, the economy and trade in Britain and well beyond.

Cameron reiterated a warning made in a newspaper interview published on Sunday that pensions and the publicly funded National Heath Service could face cuts after a vote for Brexit.

Rejecting accusations of scaremongering, he said it was his job to talk about the potential dangers of pulling out of the EU but that there was a "strong, bold, patriotic, positive case" for staying in.

"If we vote In, I think there will be actually a wall of investment," he said. "Companies that are responsible for employing people in this country and making things in this country will want to do more, employ more, make more ... if we vote out, it is a decade of uncertainty."

Britain's pound weakened by as much as 1.2 percent against the U.S. dollar immediately after an ORB poll for the Independent newspaper published on Friday showed a sharp swing toward Brexit.

JP Morgan said its analysis of four polls from the last week gave the "In" campaign a 5.1 percentage point lead, which it expected to widen in the final two weeks of the campaign.

"Based on historical experience, we think status quo bias should begin to push the polls in favour of 'Remain'," the bank said in a note to clients on Sunday.

Nigel Farage, the leader of the anti-EU UK Independence Party, described the "In" camp's warnings over Britain's economy - Europe's second biggest - as a "daily prophet of doom".

"These are ludicrous scare stories that are being put up. Even if sterling were to fall a few percentage points after Brexit, so what? The point is we have a floating currency and it will be good for exports," he told the BBC.

Farage, who has previously said the issue of Britain's EU membership would not be resolved if the result is a close vote for "In", said the referendum was likely to be the only chance to leave.

"If the Leave side was to narrowly lose, the chances of parliament giving us another referendum in the short term is probably pretty slim, so I do view this as the one great opportunity," he said.

The spiritual leader of the Church of England, Archbishop of Canterbury Justin Welby, said on Sunday he would vote in favour of Britain remaining in the EU, hailing the bloc as a force for peace and reconciliation.

(Changes paragraph 11 to read '5.1 percentage point')

(Additional reporting by Estelle Shirbon; Editing by Mark Heinrich)