Savage, who began his career with the Red Devils, thinks only one thing can ease that pressure - signing Gareth Bale.
Moyes admitted on Thursday he struggled to fathom how United had been given matches against Chelsea, Liverpool and Manchester City in the opening four weeks of the upcoming Barclays Premier League campaign.
"I find it hard to believe that's the way the balls came out of the bag, that's for sure," the Scot said in an interview with BBC Sport on Thursday.
Savage thinks the outburst was ill-advised.
"What David should have done is look at the last five games United have. It's quite an easy run-in. I have seen Stoke's run-in, they are very, very difficult but you don't hear Mark Hughes moaning," the Welshman said.
"Maybe the pressure is getting to him a little bit."
That pressure, according to Savage, stems from United's inactivity in the transfer market this summer.
Moyes has failed in his pursuit of Cesc Fabregas and the only player who has arrived at Old Trafford since he took over is young Uruguayan full-back Guillermo Varela.
Moyes has also had to deal with speculation surrounding the future of Wayne Rooney, who has been the subject of two bids from Chelsea.
"David Moyes is a fantastic manager, but he has been let down by Manchester United," said Savage, who will be a pundit for the BBC this season.
"When Sir Alex (Ferguson) left, they should have had one or two marquee signings lined up to appease the fans. That hasn't happened and there has been a bit of nervousness. The Rooney scenario hasn't gone away either."
United are also understood to be considering a bid for Tottenham forward Bale, who was named triple player of the year last season after scoring 26 goals for the north London club.
Real Madrid are willing to break the world transfer record to sign the 24-year-old, but Savage thinks United should also go in for the former Southampton player.
"If I was United, I would offer £85million for Bale," the former Wales midfielder said.
"And if they kept Rooney that would almost certainly hand United the title. It's a no-brainer for me.
"With Bale, Robin van Persie and Rooney playing together in the same team, you have to fancy United for the Champions League as well."
Spurs boss Andre Villas-Boas is determined to hold on to his star player, but Savage thinks his compatriot has outgrown the London club, who finished fifth last season.
Savage added: "Spurs aren't in the Champions League so Bale should take his opportunity. What if he has a bad season? What if he gets injured?
"The chance to sign for a big club doesn't come every day. Spurs have bought very well this summer, but they are not going to challenge for the title."