Giggs, 40, United's most decorated player, was left out of the squad for Sunday's miserable 3-0 defeat to Liverpool.
On Monday, rumours of a rift began to surface with some claims even suggesting the two clashed after the match.
However, the club have been quick to deny there is any problem at all between Moyes and the veteran Welsh midfielder.
A club spokesman is quoted by The Sun as saying: "The story of any problems between Ryan and the manager are completely false."
Giggs, who is under contract as a player, was appointed to United's coaching staff by Moyes in the summer.
Moyes is facing a huge week with Olympiakos due at Old Trafford on Wednesday for the second leg of their Champions League last 16 tie with United trailing 2-0 from the first leg in Greece.
That is followed by a trip to West Ham on Saturday followed by the visit of Manchester City next Tuesday with all matches live on Sky Sports.
Today's newspapers are full of speculation that Moyes is walking a tightrope at Old Trafford with The Sun saying the former Everton boss has three games to save his job.
The paper claims the club's board have grown increasingly frustrated with United's displays and are demanding a huge improvement - starting with the game against Olympiakos.
The Daily Telegraph says Moyes' future lays in the hands of the club's supporters who have continued to display the patience demanded of them by Sir Alex Ferguson on his departure from Old Trafford last year.
Moyes has urged the fans to get behind the team again on Wednesday and told the club's official website: "The players are capable of turning it around. We're all desperate to put things right and make sure we play better to give the supporters here something to shout about.
"They shouted [against Liverpool] and we didn't give them anything on the field. When we went to Greece, the Olympiakos fans played their part so hopefully our fans will roar us on.
"We have to go for the throat to get the win. We won't be gung-ho but we will certainly go with a mindset to overhaul the deficit as quickly as we can.
"We won't be reckless as we could find ourselves with a bigger mountain to climb but we can be forceful. We have to be if we're to keep ourselves in the Champions League."