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Giggs, appointed as a first-team coach by Moyes last summer, has not started for United since the 2-0 win against Cardiff City in late January and was not even on the substitutes' bench for Liverpool's visit on Sunday.
The 40-year-old has made just one substitute appearance in just under two months and is "fuming" about the rumour-mongering just days before United's Champions League round-of-16 second leg with Olympiakos.
Two United fanzines both ran threads on their websites detailing the alleged incident on Monday, however the club have swiftly quashed suggestions of a spectacular dressing room bust-up.
"Ryan is absolutely fuming over suggestions of a falling-out with the manager," a United spokesman said.
Giggs's experience could be invaluable should he return to the first-team fold for the Olympiakos tie on Wednesday. United were criticised for their immobility and lack of creativity in midfield again at the weekend, and Giggs excelled against Real Madrid at the same stage of the competition last year.
United's most decorated player, Giggs is likely to end his playing career at the end of the current campaign.