Jean Claude Juncker, as European Commission president, will oversee the process, while Barnier and his taskforce of negotiators deal with the United Kingdom government for the next two years.
Q&A: What Comes Next Once Article 50 Is Triggered?
Once Article 50 is triggered, Council President Donald Tusk is expected to provide an initial response to the Prime Minister within 48 hours, and the remaining European Union 27 countries will then put together a timetable for negotiations on the UK's withdrawal to take place.
Mrs May is expected to make a statement to the House of Commons on Wednesday shortly after invoking Article 50, setting out her aims.
"Everything is ready on this side", Margaritis Schinas, the spokesman for European Commission chief Jean-Claude Juncker, told a briefing.
"President Tusk has said he expected there to be an initial response within 48 hours".
A spokesman said the government wants negotiations to start as soon as possible but added that they "fully appreciate it is right that the other 27 European Union states have time to agree their position". The period can only be made longer by way of a unanimous vote of all European countries' governments.
Speaking in Swansea on Monday, during the first of a series of visits around the United Kingdom before she triggers Article 50, Mrs May said she was intent on "delivering on Brexit and getting the right deal".
Camera Catches 2-Year-Old Twins Having Epic Party After Bedtime
Nearly from the moment dad leaves the room they scamper out of their cribs and are once again fast at it. They eventually went to sleep later in the night.
The EU has yet to set a date for a summit to respond to Britain's notice of withdrawal but it should be between four and six weeks after March 29, an EU source said on Monday.
David Davis, Britain's Brexit secretary called the departure from the European Union "the most important negotiation for this country in a generation".
Formal talks between Britain and the EU must then wait for member states to approve more detailed negotiating rules and give an official mandate to European Commission Brexit negotiator Michel Barnier, officials said.
Liberal Democrat leader Tim Farron said: "Theresa May is embarking on an extreme and divisive Brexit". She has rushed this through without a plan, and without a clue.
Due to trigger Article 50, her office has said May will be visiting Wales, Scotland and Northern Ireland to hear the government's differing views on Brexit. The German defence ministry confirmed to The Financial Times it was working on joint projects.
The UK now pays about 12 per cent of the EU's budget, which means its withdrawal will impose bigger financial burdens on the likes of Germany and France.
Up to early 2019, Britain will remain a full European Union member and nothing will change beyond the exclusion of its representatives from some European Union meetings where the approach of the remaining 27 to Brexit is discussed. Reports of greater military co-operation between Berlin and Germany come after Donald Trump accused Germany of owing North Atlantic Treaty Organisation "vast sums of money".