* Odds on Brexit lengthen, sterling rises after Obama visit
* "Leave" campaigners fight back, raise immigration concerns
* But interior minister May says Britain safer within EU
* Graphic on odds: http://reut.rs/1VEth0y (Adds details on odds, Home Secretary Theresa May quotes)
By Estelle Shirbon
LONDON, April 25 (Reuters) - Bookmakers' odds have shifted sharply towards Britain voting to remain in the European Union in a referendum in June, a move in sentiment also reflected by the pound rising on Monday to a near six-week high against the euro.
Both moves followed a direct intervention by U.S (Other OTC: UBGXF - news) . President Barack Obama in favour of Britain staying in, but those advocating leaving sought to undermine his arguments and warned the "In" camp not to celebrate too early.
Obama said Britain would find itself "in the back of the queue" for a trade deal with the United States if it voted to quit the EU on June 23, and said it would be safer, more prosperous and more influential if it stayed in.
Following his intervention, which was more forceful than had been expected, the implied probability of a vote to remain in the 28-nation bloc rose to 74 percent, a jump of more than 10 percentage points compared with a week ago, according to odds from bookmaker Betfair.
"'Remain' was already a strong favourite before Barack Obama's visit, but his comments sparked another wave of trading over the weekend with 'Remain' now backed into 1/3," said James Midmer, a spokesman for Betfair.
"The market could eclipse Betfair's record for traded volume for a single political market; 3.5 million pounds ($5 million) has been matched in the last week alone," he said.
The probability of a "Remain", or "In", vote implied by the odds was at its highest level since September 2015, Betfair data showed.
The pound traded near a six-week high against the euro, with traders saying Obama's intervention was helping sentiment as it underlined the weight of argument from global and financial leaders in favour of the "Remain" camp.
"CROWING TOO SOON"
The EU issue has split the ruling Conservative Party, with Prime Minister David Cameron leading the campaign to stay in while six of his cabinet ministers and a large proportion of his party's lawmakers are openly campaigning for Brexit.
The "Leave" campaign fought back on Monday, with figurehead Boris Johnson, the outgoing Conservative mayor of London seen as a potential successor to Cameron, saying Obama had "bullied" British voters and that his warning on trade was "ridiculous".
In an opinion column in the Daily Telegraph newspaper, Johnson accused the "Remain" camp of "crowing too soon".
Another prominent "Leave" campaigner, Justice Secretary Michael Gove, wrote in the Times newspaper that the next wave of EU expansion, which could include Turkey and Albania, would create a migration "free-for-all".
But Home Secretary Theresa May, the interior minister who is in the "Remain" camp, said "nobody should think leaving the EU is a silver bullet that is suddenly going to solve all our immigration problems".
She (Munich: SOQ.MU - news) argued in a speech that controlling immigration and preventing criminals and terrorists from moving freely around the bloc were challenges best tackled by constant cooperation with EU partners and continual refining of the rules.
"Leaving the EU would not mean we could just close ourselves off to the world ... Most of the international terrorism casework that crosses my desk involves countries beyond Europe's borders," May said.
"So my judgment as home secretary is that remaining a member of the European Union means we will be more secure from crime and terrorism." ($1 = 0.6916 pounds) (Editing by Guy Faulconbridge/Jeremy Gaunt)