EASA license training could not be done anywhere in the world,
The training has to take place to a certain extent in an EU country, and the IRT itself administered in an EU country.
But there are EASA flight schools in the US ! How can they operate then ?
They have specific approvals from some EU countries. I think historically it was usually the UK which ran these. @tumbleweed might know more.
The history is a long one dating to the days of the the Arnold Scheme under which seven British Flying Training Schools were set up in the US at Terrel TX, Lancaster CA, Miami OK, Mesa AZ, Clewiston FL, Ponca City OK, and Sweetwater TX. Other schools were created elsewhere in the empire and commonwealth—in Australia, Canada, New Zealand, South Africa, and Southern Rhodesia—under the Empire Air Training Scheme, Joint Air Training Scheme (South Africa), and British Commonwealth Air Training Plan. There’s some history about the BFTS here, http://www.5bfts.org.uk/articles/history/fullhistory.aspx
While those plans died out with the end of the war they must have paved the way for British civilian flying training in the late 1980s or early 1990s. To the best of my knowledge the first person to organise UK CAA approval for a US-based training school was Stephen Fisher at Flight Safety in Vero Beach, FL. He was followed by others like Ormond Beach Aviation set up by Adrian Thompson around 1992 if memory serves, and another at Kissimmee Municipal, now Kissimmee Gateway, managed by the Cranfield-based Cabair International, and Naples Air Center originally run by Nikki and Richard Gentil. One school running under IAA approval was a satellite of Waterford-based Pilot Training College. When it went into administration the training in Florida was reorganised by Florida Institute of Technology to be done directly under EASA oversight. FIT was the first organisation to achieve this feat at a cost of six to seven figures. I vaguely recall Sabena having a school in the US and it must have been approved under the Belgian CAA/DGAC.
Today training organisations providing EASA training outside the member states may still come under national approval if operating as satellite bases. Naples Air Center operates on this basis as a satellite of Andrewsfield Aviation in Essex. Otherwise enforcement and oversight must be done by EASA directly. The basis for this is article 78(1) of the Basic Regulation.
The US schools can train up to the CPL, but the IR needs specific time in the EU.
They can also be approved to train for the IR but acclimatisation flying and the skill test must be done in one or more EASA member states.