No. All flights across the Atlantic are routed onto specific tracks that are drawn-up each day to (going eastbound) take advantage of tailwinds and (going westbound) to avoid the worst of the headwinds. Before every flight, the airlines' dispatcher will review that evening's NAT tracks and select the best one for each flight (least flight time and fuel usage) as well as the ideal altitude and speed and will file that in the flight plan. Once in flight, the pilots must contact the Oceanic Control Center for final clearance. There is no guarantee that a particular flight will get its desired routing as all aircraft want to operate in the same naorrow band of altitudes and on the most direct routing. It's up to the controllers to sort it out.

There are four Oceanic Control Centers covering the Atlantic. They essentially have divided it up into four quadrents, each with responsibility for one quadrent. The dividing lines are at 30 Degrees West Longitude and, I think, 42 or 45 Degrees North Latitude.

The four Oceanic Control Centers are:

1 - Gander Oceanic, Newfoundland Canada (controls the northeast quadrent)
2 - Shanwick Oceanic, a combination of Shannon Ireland and Prestwick, Scotland, with the radio operators in Shannon and the controlers in Prestwick (controls the northwest quadrent)
3 - New York Oceanic, Long Island (controls the southeast quadrent)
4 - Santa Maria Oceanic, Azores Islands, Portugal (controls the southwest quadrent)

A typical U.S./Canada to Europe flight will controlled by Gander and Shanwick. Flights from Europe to the Carribbean and South America are more likely to be controlled by New York and Santa Maria.