West Ham smashed Lyon 3-0 at the Groupama Stadium in France to go through to the semi-finals of the Europa League on Thursday night.
The Hammers dominated the game in Lyon and will now face Eintracht Frankfurt for a place in the final of the competition this season.
With Barcelona out of the tournament, many are anointing West Ham as the favourites to win the trophy and Osman conceded that puts extra pressure on David Moyes’ side.
He pointed out that West Ham have been enjoying their journey in Europe but will now have to deal with the pressure of actually going all the way and winning the competition.
Osman said on BBC Radio 5 live: “They’ve got a different issue coming up now. Up until this point it was a let’s see what they can do, let’s enjoy the European trip.
“I think with Barcelona going out, suddenly this competition has opened up for them. Suddenly they could potentially win this, it’s a serious trophy that they could win.
“You could potentially even call them favourites at this point.
“And that brings a different kind of pressure than they’ve had up until this point.
“It’s something else they’ve got to now learn to deal with.”
The former Premier League star stressed that West Ham made a statement through the performance against Lyon and Europe will not take note of the Hammers.
He believes if they manage to win the trophy it could transform them given they would be in next season’s Champions League and would be able to attract a higher calibre of players.
“This is a performance the whole of Europe will be taking a look at.
“They’ll be noticing, well they beat Sevilla comfortably in the last round.
“They’ve just gone to Lyon who are a Champions League team and won that game.
“These kind of performances can put you on the map.
“If you can go on and win this competition then suddenly you can strengthen in the summer.
“Players will want to join this team, its a team that are progressing, it’s a team that is building towards something.
“It’s a team that is trying to bring European success to the club.
“It’s a huge statement performance from West Ham.”
West Ham could be tested with bids for Declan Rice in the summer and winning the Europa League could potentially convince the midfielder to stay.
