The 2-0 defeat at fellow strugglers West Ham was a ninth loss in 12 Premier League matches, from which the Toffees have taken just five points, and left them second-bottom in the table.
Moshiri, attending his first match in 15 months, was asked after the game at the London Stadium whether it was time for Lampard to be sacked after almost a year at the club.
“I can’t comment. It’s not my decision,” he told Sky Sports.
The response was met with bewilderment from fans and former players alike, especially after a couple of weeks ago Moshiri had said he has sacked previous managers off the back of adverse reaction from supporters.
Former defender Alan Stubbs, who has been an outspoken critic of the current situation at his old club, wrote on Twitter: “It’s not my decision” Who’s is it then, enlighten us Agents, his mate, kit man’s, chef, media??????”
Ex-Toffees striker Campbell replied to his former team-mate: “I can’t believe all this foolishness! We all know the buck stops with the owner Stubbsy!”
Irrespective of the owner’s apparent inertia, Lampard’s job continues to hang by a thread as a first relegation since 1951 looms large.
The former Derby and Chelsea boss has won just nine and lost 21 of his 38 matches since taking over from Rafael Benitez at the end of January 2022.
Since the start of last year, Everton have lost seven of their nine league games against teams starting the day in the relegation zone, including each of the last four.
Lampard has said previously he did not expect the club to be much better off than last season when they managed to avoid relegation in the penultimate game of the campaign.
After defeat to West Ham he pointed to the gradual decline at Goodison since 2015 and stressed that, especially now working under severe financial constraints, there was no quick fix.
“It’s a big job. I knew that when I came in as we were already looking at a relegation battle,” he told BBC Sport.
“I don’t think when you’ve made steps downwards over the years it is a case of being able to turn that instantly, that’s naive.
“History moves and there is nothing in this league which gives you a right to do anything. We have seen huge football clubs go down and come back up.
“It is all about the moment and how you try to move forward as a club and if you don’t do that then you will come into tough times.
“It’s maybe normal we are fighting again. It is impossible for me to jump into the future. There a long time to go this season and we have to try to turn the corner.”