I think it is not likely, and I am sure it will be bad decision for them if they decide to.
Legislation do not allow them to own more then 49% in EU carrier and they probably will not get wide rights to operate routes inside EU with their own
AOC.
But legislation is one thing, attractiveness for the passenger and business sense is much more stronger. Turkish Airlines have so much connecting passengers to North America from Middle East, India, Iran, Africa, etc. No look at the passenger let's say from Mumbai, who needs to get to let's say to Miami. Options - one connection stop on ME
B3 (sooner or later they will expand to almost all US cities), one connection on European carrier (BA,
LH, KLM,
AF, others).
TK? Two connections in
IST and
CPH? Makes no sense. You can win some passengers on price, but they will be lowest yield "price bargainers" (and some aviation enthusiasts, who love take-offs, lounges and landings).
What I heard, Jet Airways hub in Europe do not work so well. And they are in better position - they have home market India splitted to few metropolis and each of them cannot sustain direct flights to US. So if you are traveller in Delhi, you need to transfer let's say in Mumbai (that is one stop) or the same one stop in
BRU/
LHR or anywhere else, what even make more sense to stretch legs in the middle of the journey.
I don't tink
TK hub in EU could make sense. Very small benefits for the passengers, strong legal barriers and try to enter the market with the worse product (two stops will never win vs one-stop).