United bounced back impressively from the 6-1 defeat to Tottenham before the international break, with three wins and a draw in all competitions.
But that mini-improvement was derailed again when Arsenal earned their first win at Old Trafford since 2006 thanks to a penalty from captain Pierre-Emerick Aubameyang.
Questions are once again being asked of Solskjaer with Mauricio Pochettino potentially waiting in the wings to take over.
And McClaren, who worked under Sir Alex Ferguson, believes Solskjaer's position will come under threat unless greater consistency is found.
“What is probably annoying most people, it’s certainly annoying me, is the consistency of the players is not there – that’s why they’re getting inconsistent results," he told talkSPORT.
“Two weeks ago people were calling for Ole‘s head, then last week after the 5-0 against Leipzig it was all ‘what a manager’, ‘stick by him’.
“I’ve got to say, the players have got to take responsibility. I know you need consistency of shape, but you need consistency of performance from players, and they’re just…
“One week they’re a nine or ten, the next week they’re four and five, and a top team can’t have that.
“But both the players and the manager have got a responsibility, really. You’ve got to get the best out of your players and make sure that’s week in, week out, otherwise Ole and every manager knows you’re going to be in trouble.
“At Manchester United you have to win, every defeat is a crisis.
“You might get out of the mire with a 5-0 against Leipzig, but you know on a Thursday morning you’ve got to win on Sunday, or you’ll be back to being criticised.
“That’s how it works at big clubs. You get beat once and it’s like a morgue, it’s like a crisis. The players have to respond, but they’re not responding."