United beat Everton in the early kick-off on Saturday after losing to Istanbul Basaksehir in the Champions League on Wednesday with Solskjaer saying the quick turnaround time was an "absolute shambles."
In contrast, Liverpool and Manchester City, who both played in the Champions League on Tuesday had a late afternoon kick-off on Sunday.
"I have spoken to the Premier League," Klopp said after his side's 1-1 draw with City. "No team on a Wednesday night should have the 12.30 BST kick-off on Saturday.
"Okay, Sunday, no problem, not the 12.30 BST because you wake up and play football. This is the recovery phase. Premier League has to change, Sky, BT everyone has to talk to each other."
Solskjaer added that the quick turnaround led to injuries for his players.
"The authorities set us up to fail, who is responsible? We've had enough of that. Players this season, in these times, a pandemic, mentally and physically it's draining. Let us play on a Sunday, there's an international break after this, it's a joke."
United have only won three of their opening eight Premier League games this season and back-to-back defeats against Arsenal -- in Solskjaer's 100th game in charge of the club -- and Istanbul Basaksehir led to speculation about his future at Old Trafford.
After the match, Liverpool defender Andrew Robertson said: "When you come to Man City against a team like them, managed by Pep [Guardiola], a draw is a good result.
"We have struggled here in the league the last couple of season so we can't be too unhappy with a draw. But we feel that if we had had a bit more composure in the last third, we could have won it."
The Scotland international added about Alexander-Arnold's injury: "We have been relentless this season. Since the last international break, we have played Saturday, Tuesday, Saturday, Wednesday. It has been a game every two or three days.
"Unfortunately, muscle injuries are going to happen. It looks like Trent has done one, hopefully it is not too bad. When you start playing and you feel something, it is never a good sign. Fingers crossed for him."