Both ESPN and the Guardian are suggesting that the underfire boss needs to improve results quickly if he is to be kept in his job.
Their report says that the Glazer family are still behind Moyes but they "are now open to the idea of making a change and the next three games could prove decisive."
The Guardian's report takes a similar line saying: "United's Champions League last-16 meeting with Olympiacos on Wednesday evening has become vital to the manager's long-term job prospects."
After that Champions League clash, United play West Ham United at Upton Park and then the Manchester derby at Old Trafford next Tuesday.
Moyes admitted on Sunday he is finding succeeding Sir Alex Ferguson a more difficult job than he anticipated as a 3-0 home defeat to Liverpool dealt another blow to their season.
This season United have won just once against clubs in the top nine and Moyes said that was obviously something that had to change.
"That tells you we're not doing as well as we should be. We have to play better and make ourselves harder to beat, harder to play against and we are also going to have to make sure we are creating and taking more opportunities," he said.
"We have a lot to do to do both those things. I think the job was always going to be hard. Yes, it is harder (than I thought) I would say so."
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