* May needs pact with N.Irish party for slim majority
* Britain's talks on exiting EU to start on Monday
* N.Irish DUP leader wants "sensible Brexit"
* Brexit priority should be jobs, growth -Hammond
By William James and Padraic Halpin
LONDON/DUBLIN June 16 (Reuters) - Britain is likely to go into arduous talks on its exit from the European Union without a deal to keep Prime Minister Theresa May in power as negotiations with a Northern Irish "kingmaker" party grind into a second week.
After a tumultuous week that pitched Britain into its deepest political crisis since the Brexit referendum a year ago, May's future was uncertain due to her botched gamble on a snap election and a muted response to a deadly fire in London.
Fighting for her political survival, May has been trying to strike a deal with a small Northern Irish Protestant party to avoid a second election that could delay Brexit talks and damage the $2.5 trillion economy.
While she is ultimately expected to reach a deal, a source in the Democratic Unionist Party (DUP) told Reuters that an agreement to support her minority government was not likely before next week. The price of such a deal remains unclear.
"If there is the expected positive outcome, it will be at least the start of next week before anything is signed off," a DUP source told Reuters.
Having lost a majority in parliament, May needs the DUP's 10 lawmakers to win any kind of vote, including on the pieces of legislation needed to enact Britain's divorce from the EU.
When asked if a deal was in the making, DUP leader Arlene Foster told reporters in Dublin: "I think you know it takes two to tango and we're ready to dance."
A source in May's Conservative Party said talks continued on Friday. The party has refused to give a time frame for reaching a deal, though May is due in Brussels for an EU summit on June 22-23 when she will want to show she has a solid grip on power.
Besides forging a deal to keep her job and preparing for Brexit talks, May is also grappling with a crisis over the breakdown in power-sharing between the pro-British DUP and the Catholic nationalist Sinn Fein party in Northern Ireland's separate, devolved government.
She is also trying to contain outrage at home over a London tower block fire which left at least 30 people dead.
After facing criticism for not meeting survivors on Thursday, May visited a hospital on Friday to speak to some of the people who escaped the fire.
While European leaders try to gauge what to expect from the Brexit talks due to begin in Brussels on Monday, May is so weakened that her own Brexit strategy is the subject of public debate in her own party, and by her potential allies.
"We want to see a Brexit that works for everybody, not just in Northern Ireland from my perspective but in the Republic of Ireland as well, so it is about a sensible Brexit," DUP leader Foster told reporters.
May wants to negotiate the divorce and the future trading relationship with the EU before Britain leaves in March 2019, followed by what she calls a phased implementation process to give business time to prepare for the impact of the divorce.
But her vulnerability at home means that her plan to leave the EU's lucrative single market and customs union in order to impose strict limits on immigration is under intense scrutiny.
Prompted by her poor election showing, particularly among pro-EU young people who fear losses of jobs and opportunity for Brexit, some of her most senior ministers and two former Conservative prime ministers have called for a rethink.
May's finance minister, Philip Hammond, said on Friday that Britain should prioritise protecting jobs and economic growth.
"My clear view, and I believe the view of the majority of people in Britain, is that we should prioritise protecting jobs, protecting economic growth, protecting prosperity as we enter those negotiations and take them forward," Hammond said.
"As we enter negotiations next week, we will do so in a spirit of cooperation, taking a pragmatic approach, trying to find a solution that works both for the UK and the European Union 27," he told reporters before a meeting of the 28 EU finance ministers in Luxembourg.
Britain's Brexit ministry said on Friday that no deal could be struck on exiting unless the future relationship with the bloc was taken into account.
"Our view is that a withdrawal agreement and terms of the future relationship must be agreed alongside each other," a spokesman for Britain's Brexit ministry said.
"We believe that the withdrawal process cannot be concluded without the future relationship also being taken into account. The withdrawal and future are intimately linked."
EU leaders have given chief Brussels negotiator Michel Barnier no authority to so much as talk about future trade with Britain until he clinches outline deals on Brussels' priority issues, including London's exit bill.
Addressing her party on Monday after one of its most memorable electoral failures, May said she would take a broader, more consultative approach to the Brexit talks.
But after a generation of discord over Europe inside her party, May's future could depend on her ability to please both the eurosceptic and pro-European factions in her party. (Additional reporting by Amanda Ferguson; writing by Guy Faulconbridge; editing by Mark Heinrich)