* Davis and Barnier set to launch first week of talks Monday
* EU pressing London to agree "Brexit bill" in principle
* Citizens' rights, borders, Ireland also on agenda
* Britain will be out of EU, deal or no deal, in March 2019
By Alastair Macdonald
BRUSSELS, July 16 (Reuters) - EU officials hope the British government shows more urgency about a Brexit deal when its negotiators come to Brussels on Monday for a first full round of talks aimed at smoothing Britain's departure.
"The hard work starts now," European Union chief negotiator Michel Barnier said on Wednesday, again sounding a note of alarm that London has yet to provide detailed proposals on a range of key issues, with barely a year left for bargaining.
A year after the referendum vote to leave, Prime Minister Theresa May still faces a complex task in finding consensus at home on what kind of Brexit Britain wants. That job was made all the harder by losing her parliamentary majority last month.
Her Brexit minister, veteran anti-EU campaigner David Davis, will meet Barnier, a French former cabinet minister, at the European Commission's Berlaymont headquarters at 9:15 a.m. (0715 GMT) on Monday for a brief public handshake before formal business begins.
Their teams will spend most of the next four days in smaller working groups, trying to identify areas of accord and discord on a set of issues agreed on during an initial day of talks on June 19. On Thursday afternoon, Davis and Barnier should hold a news conference, to fix publicly what has been agreed so far.
The priorities, notably rights for expatriate citizens, how much Britain may owe to the EU budget and how to manage the new EU-UK border, especially with Ireland, are ones both sides want to settle in a withdrawal treaty. Barnier says this must be ready by about October next year if it is to be ratified on both sides of the Channel before Britain leaves in March 2019.
"The clock is ticking," he said on Wednesday, displaying a degree of impatience with British ministers who continue to dismiss EU demands that they first must agree in principle that London will owe the Union a hefty amount -- probably in the tens of billions of euros -- to cover its existing commitments.
"The first serious test of the negotiations will be them agreeing to pay the bill," a senior EU official said -- the coming week is a vital moment to establish rapport among the senior civil servants who will handle what is arguably the most convoluted and far-reaching diplomatic deal of modern times.
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Without "significant progress" on all three priorities in the divorce, Barnier warns, EU leaders will not let Davis open talks on a free trade relationship, which May and much of British business want to have ready by the time Britain leaves.