The Hammers are currently sat in third place in the Premier League table, and Manchester City are the only team in England's top-tier to have hit the back of the net on more occasions.
Indeed, Keown feels that West Ham could have been sitting at the top of the league if they had taken points against the likes of Bournemouth and Norwich City at home.
“With West Ham this season, it’s a bit like Southampton last year – we keep waiting for them to drop off,” he said. “Eventually it happened to Southampton but West Ham are still going so strong. When you look at their performances, particularly away from home, you can’t help but think what might have been if they had won the games they would have expected to.
"They beat Arsenal, Man City, Liverpool all away and Chelsea at home. If they had beaten Leicester, Bournemouth or Norwich at home – as they might have expected to – they would be top of the league, which is an incredible thought. If people hadn’t already, then it’s time to start taking West Ham very seriously.”
Despite West Ham’s brilliant start, Keown says that the worry now is that they will become complacent.
The former Arsenal defender is fearful that the Hammers could become over-confident, but he believes that Slaven Bilic will ensure that they don’t rest on their laurels.
“The worry now is that they might take their eyes off the ball and drop their level," he added. "They can’t afford to get complacent but with Slaven Bilic in charge I don’t expect them to. He has made an excellent start but the club has better players this year, with Dimitri Payet the standout, and the likes of Carl Jenkinson and Aaron Cresswell are another year on with their development.
“We are not talking about Adrian as much as last season because he is not having to make so many saves and the squad spirit is soaring. If they can keep it up they can certainly challenge for Europe.”
West Ham are currently unbeaten in their last seven Premier League matches, and they were victorious over Chelsea in their most recent outing.
Bilic’s men’s next match comes against Watford, and a victory in the contest could see them rise to the top of the Premier League table, if other results go in their favour.