Red Bull announced on Wednesday morning that Fallows would become its new aero chief, and that Peter Prodromou, who has agreed a deal to join McLaren, would begin a period of "gardening leave" before taking up his new post at the Woking team.
Fallows was formerly a team leader in Red Bull's aerodynamics division.
He left the squad for a period last year before the announcement that he would return in the lead role.
But prior to that, Fallows had also been tipped for a move to McLaren.
A spokesman for McLaren told AUTOSPORT that the team believed it had secured Fallows' services and had referred the situation to its legal team.
"Dan Fallows has a legally binding contract with McLaren, and the matter is now in the hands of our lawyers," said the spokesman.
McLaren has undertaken a number of staff changes in recent months as it bids to recover from a disappointing 2013 season.
Long-time team boss Ron Dennis has reclaimed direct control of the F1 operation and ex-Lotus chief Eric Boullier has come in as racing director, with erstwhile team principal Martin Whitmarsh losing his position.
Neither McLaren nor Red Bull has won a grand prix so far in a 2014 season dominated by Mercedes to date.
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