* May renowned for keeping bankers at arm's length
* But British prime minister sits down with JPMorgan boss
* Banks sense more sympathetic hearing on Brexit fears
* Brexit negotiator promises City "very much in my mind"
LONDON, Nov 20 (Reuters) - A few hours before Theresa May lost her second cabinet member in a week, the British prime minister found time amidst the crisis to meet one of the world's most powerful bankers.
With May renowned for keeping finance chiefs at arm's length, some commercial rivals suspected special treatment in her willingness to sit down with Jamie Dimon, chief executive of U.S. investment bank JPMorgan.
Others, however, sense a subtle but significant change in the political climate with May listening more to the worries of an industry struggling to fathom out Britain's departure from the European Union.
Her talks with Dimon on Nov. 8 raised eyebrows - and some hackles - around the City of London financial center.
"It's a bit outrageous," said a senior executive at one of Britain's top banks. "What other bank CEO is getting as much one-on-one face time with the prime minister? We certainly aren't."
From Margaret Thatcher in the 1980s to David Cameron last year, British prime ministers have traditionally held the door open for leaders of a finance industry that pays more than 70 billion pounds ($92 billion) in UK tax every year and employs more than a million Britons.
Not so May, who has appeared reluctant to meet bank executives. Still, Dimon secured the face-to-face talks between the abrupt departures of her defense and international development ministers which rocked the minority government.
Shortly before last year's Brexit referendum, Dimon warned that 4,000 out of JPMorgan's 16,000 British jobs might have to move abroad to ensure the bank could continue selling services across the remaining nations of the EU.
Dimon has since softened his tone, but the British bank executive asked whether the original estimate had won JPMorgan the access. "I don't know if it's because they've threatened to move the most jobs - maybe we should say we're moving 4,000 too," he said.
But something is afoot. Reuters has spoken to more than a dozen executives at large banks and government officials, with many saying May is showing greater willingness to press the City's cause in Brexit negotiations.
In return, the bankers are scaling back their plans to move jobs to rival financial centers such as Frankfurt, Paris and Dublin, at least during any transitional period after Britain leaves the EU in March 2019.
Britain and the EU have yet to begin discussing financial services at the Brexit talks, worrying bankers who fear the government has left it too late to negotiate a transition period or secure a good deal.
"Our view is that the government frankly is in chaos," said another senior executive at a U.S. bank. "We are really nervous."
However, May outlined to Dimon her vision for a transition period after Brexit day - March 29, 2019 - and a future trade deal with the EU, easing concerns that Britain may crash out of the world's biggest trading bloc without any agreement.
A JPMorgan spokeswoman expressed cautious optimism after the Downing Street talks.
"While some uncertainty remains, we feel that the government understands the concerns of international firms such as ours, and the economy more broadly," she said. "We were grateful for better clarity on the government's objectives in the Brexit negotiations."
A spokesman for the prime minister did not respond to requests for comment.
However, sources with knowledge of the meeting noted that she can make promises on her intentions for the Brexit talks but not on their outcome. "We got clarity on what they're looking for, not on how it will unfold," said the source.
The bankers pressed their case for a period of adjustment to the post-Brexit financial services regime, rather than an abrupt change as soon as Britain leaves.
"We're all looking for transition and a good future deal," said the source.
In return, the JPMorgan team indicated there would be no major staff upheaval, at least on Brexit day.
"They told her we will try to move as few people as possible on Day One. It will be a few hundred people and we will keep as many people as possible in the UK," said another source.
For the banks, moving top London employees to the continent is expensive and unpopular with staff reluctant to uproot their families. But longer term, a significant number of jobs may still shift if a final deal is not to their liking.
CITY CHARM OFFENSIVE
As in a number of countries, banks have been deeply unpopular in Britain since the global financial crisis. But their leaders felt particularly unloved after May came to power last year and cooled what critics regarded as an overly cozy relationship between the government and big business.
May confined her contacts with bankers largely to round table gatherings such as at Davos in January, a meeting Dimon also attended. Some Wall Street bosses were left expressing frustration with a lack of detail on the Brexit process, and questioning May's commitment to protecting the finance industry
The relationship remains measured. May was scheduled to make only a five minute courtesy appearance at the talks with Dimon, hosted by finance minister Philip Hammond. In the event she stayed a quarter of an hour.
Still, those who connect government and business have noticed a change.
Iain Anderson, executive chairman of public affairs company Cicero, said a thaw began after the May lost her parliamentary majority in a June election and the government started asking companies for more input on the Brexit talks.
"The mood has improved from last year when I don't think I have ever seen businesses as emotional and angry," said Anderson, whose firm represents many FTSE 100 companies.
Government officials confirmed May's change of tack. "She is far more skeptical about engaging with business than her predecessors," said one official, who declined to be named.
"This created bitterness because people are getting less face time," said the official, but added: "The situation has improved since the election."
Brexit minister David Davis, whose talks with chief EU negotiator Michel Barnier appear deadlocked, has led his own charm offensive in the past week or so.
Davis said on Tuesday the financial industry would be exempt from any curbs on immigration from the remaining EU countries. That is likely to contrast with tough new rules planned for arrivals of lower-skilled workers.
In another speech, he emphasized the importance of an industry which accounts for about 12 percent of Britain's economic output.
"When I sit round the negotiating table with Michel Barnier, or round the cabinet table with my colleagues, it's always with the importance of the City very much in my mind," Davis said.
Only a year ago, he was accusing bankers of leading a "blame Brexit festival" and lying about the number of UK employees they would have to fire.
Davis had surprised executives in his initial meetings by telling them they would not get any special favors in the negotiations, according to people who attended.
The head of one of Britain's largest finance companies, who has met Davis several times in the last year, said the minister has accepted the damage to the wider economy if firms go overseas.
"Davis has got the message," the executive said.
According to government officials, the atmosphere has improved since the banks scaled back their estimates of how many jobs will move to more realistic levels.
"There has been an effort on both sides to improve the relationship," he said.
($1 = 0.7580 pounds) (Editing by David Stamp and Guy Faulconbridge)