McLaren, insisting Fallows has a "legally binding contract".
But Red Bull on Wednesday said Fallows has in fact become the reigning world champions' new 'head of aerodynamics'.
"Previously working as a (aerodynamics) team leader, he left the team for a short time last year," Red Bull said in a statement.
"He has now returned ... to begin working in his new role," the team added.
It is believed Red Bull convinced Fallows to stay by offering him the departing Prodromou's job and salary.
"The matter is now in the hands of our lawyers," team boss Eric Boullier said in a teleconference on Wednesday.