(Recasts, adds quotes, details)

By Anirban Nag

LONDON, May 24 (Reuters) - Sterling rose to 3-1/2 month highs against the euro on Tuesday, boosted by growing support for those campaigning to stay in the European Union and as Bank of England chief Mark Carney made no further comment about the risks of a Brexit vote next month.

Speaking to lawmakers, Carney also said there was a "higher bar" in drawing conclusions about the economy due to substantial uncertainty around the Brexit referendum, noting that the BoE would be able to deal with any fallout from the vote.

He also said that it was not clear whether the bank would cut rates if the economy slowed after a possible Brexit, but said leaving the EU would reduce the probability that the next move in rates would be up. Carney said the BoE would not seek to counter moves in the exchange rate if Britain voted to leave.

Some traders said Carney sounded more measured than in previous statements on Brexit, cooling nerves in the market over his position and the extent of risks to the economy.

Sterling, already well up on the day before Carney spoke, was 0.9 percent higher at $1.4625 GBP=D4 , and rose to 76.43 pence per euro EURGBP=D4 , its strongest in more than three months. Traders said automatic stop-loss orders was a factor.

Sterling had started on a strong note on Tuesday, boosted by a poll that showed the "Remain" camp was well ahead of "Leave" rivals with just a month to go before a vote on Britain's future in the European Union.

The latest poll from ORB, published in the Telegraph newspaper, gave the "Remain" camp a 13-point lead over their "Leave" rivals, after winning support for the first time from a majority of men, those aged over 65 and Conservative voters.

"We haven't seen anything new (from the Treasury Select Committee) to change our view on the outlook for monetary policy, as much of the conversation has centred around the MPC views on the implications of the EU referendum," TD Securities European head of currency strategy, Ned Rumpeltin, said.

"Markets have also paid some attention to the referendum polls, with the latest Telegraph-ORB poll in particular showing a shift in favour of 'Remain' in some of the key demographics that had been generally supporting the 'Leave' side," he said.

Sterling's gains were also underpinned by stock market gains in Europe.

The pound tends to underperform during times of risk aversion because of the huge investment flows Britain relies on to balance its more than 5-percent current account deficit. Flows dry up when investors are worried about growth and market stability - jitters that are heightened by any developments that make a Brexit look more likely.

Brexit concerns drove the pound down 11 percent on a trade-weighted basis between mid-November and early April, when it hit a 2-1/2-year low, but it has recovered almost 5 percentage points of that. That rise came as investors priced out chances of a rate cut that some were factoring in if Britain opted to leave the union.

"We would be careful though, as a poll showing a 'Leave' advantage may be enough to push the currency off the cliff and erase some of its recent gains," IronFX Global senior analyst, Charalambos Pissouros, said.

(Editing by Louise Ireland) ((anirban.nag@thomsonreuters.com; +44 20 7542 8399 ; Reuters Messaging: anirban.nag.thomsonreuters.com@reuters.net))