The Hammers have made a flying start to their Premier League season, and they currently sit in sixth place in the top-flight standings.
However, Keys has warned West Ham fans that their season could take a similar turn to what it did last year, where they slipped down the table at an alarming rate after Christmas.
“I can hear West Ham fans screaming again, 'oh yea, what about us? What about our start?’,” he said.
“Great start, but it's too similar to last season to get carried away with. Right now the Hammers are two points - yes two points - better off than they were a year ago. Only two points.
“It's a start that's led David Sullivan to be talking about a league and cup double. Nurse - keep an eye on him. No chance. Absolutely no chance at all.
“West Ham's test will come at the turn of the year - just as it did last season.”
West Ham supporters and owners have bought into Slaven Bilic’s ethos at Upton Park, which has seen them become one of the most threatening teams in England’s top tier.
The Hammers have averaged two goals a game throughout their opening 11 league fixtures this season, and their attacking prowess has seen them claim some notable scalps.
Wins over Arsenal, Manchester City, Chelsea and Liverpool have highlighted their quality, but they are under pressure to bounce back from a disappointment after losing to Watford at the weekend.
The Hammers were outclassed by the Hornets in their most recent outing, and they will be keen to get back on track straight away when they take on Everton on Saturday.