As Sportsmail revealed last weekend, Moyes was nearly certainly taking his last game in charge of the London club when he guided them to victory against Everton on Sunday.
Moyes took over at West Ham in November with the club in the relegation zone.
"I'm told it's "touch and go" whether he stays at West Ham".
The Chilean cut his managerial teeth in South America with clubs such as San Lorenzo and River Plate before trying his luck in Europe with Villareal, Real Madrid and Malaga.More news: Marvel 'has plans' for first Muslim American superhero character
The former Portuguese central defender has been coaching since retiring as a pro at the age of just 32.
He is now in his second season with Shakhtar and has guided them to back-to-back league and cup doubles, and into the knockout stage of this season's Champions League. However, we can reveal that Sullivan has already decided that he wants a new boss at West Ham and has informed the Scot - who was only handed a short-term deal to the end of the season - his contract won't be renewed.
Shakhtar Donetsk chief Paulo Fonseca is also under consideration, along with ex-Hull and Watford man Marco Silva - who is also on Everton's radar.
West Ham came out of this battle with the points but both sets of fans know they've got a lot of work to do in the summer to stand any chance of getting to where they think they should be - in Europe.