The interim United boss led his side to a 4-0 victory over Norwich on Saturday Night Football, which led to a run of bets on him to get the position on a permanent basis and caused his odds to tumble.
Louis van Gaal remains the clear favourite for the post at 1/4 in what is increasingly looking like a two-horse race, with the likes of Carlo Ancelotti, Jurgen Klopp and Jose Mourinho all drifting out to 33/1.
Reports had emerged that a deal had been agreed for the Holland coach, who leaves his position after this summer's World Cup, to take the reins this summer.
However, the club have told Sky Sports that nothing has been signed with Van Gaal or any other candidate.
Paul Wiggins, Sky Bet's Head of Football, said: "It's hardly surprising to see a run of money on Giggs after a convincing victory live on Sky Sports, prompting us to significantly cut his odds.
"However, the visit of struggling Norwich was arguably the ideal first game in charge and we still favour the vastly more experienced Van Gaal to get the post given the enormity of the task ahead in transforming United's fortunes."