Moyes came in as a replacement for Slaven Bilic earlier in the season and initially agreed to remain in charge until the summer. Talks over his future should be underway, with the campaign set to end this weekend.
According to Sun reporter Alan Nixon the Scot has made Butland his chief target, and has already asked the club to attempt a deal for the England stopper if he keeps his job as Hammers boss.
The Potters are now headed down to the Championship and it's likely they'll lose a few players. And with Butland being one of the stars with the highest profiles in the squad, they shouldn't be surprised if they're inundated with offers in the summer.
The player himself could be keen to leave given the unfortunate development, so it's likely he's back in the Premier League next term.
West Ham fans could be a very anxious bunch this summer after one of their chairmen agreed to take a back seat as it relates to transfers following massive protests. The club did show ambition in the market last summer, though, bringing in big names like Marko Arnautovic and Chicharito. They also recruited then-free agent Patrice Evra after the close of January window.
Moyes has managed to keep the Londoners in the top flight this season. And if he does keep the job, he can expect a tougher challenge going into his first full term as boss. Signing Butland, however, would probably make things a bit easier.