The club narrowly missed out on a trip to Wembley after being held by a late goal against Manchester United at Old Trafford before losing 2-1 in the replay at Upton Park last week.
A 3-1 win over Watford last time out kept the Hammers on course for European football next season with games against West Bromwich Albion away, Swansea and Manchester United at home and Stoke City away remaining.
'It's not impossible'
"It is not impossible for the club to win the remaining four matches of the season and I know that Slaven and the players will be doing everything they can to get maximum points from those games," Gold told whufc.com.
"The European dream is still alive."
What was significant about West Ham's win last time out was that it set a new top flight record of going 15 games unbeaten at home, a run stretching back to August.
European football at the Olympic Stadium?
And it is made all the more significant by the fact this season is West Ham's last at their beloved Boleyn Ground home of 112 years before their big move to the 60,000 seater Olympic Stadium in the summer.
Should West Ham do as Gold predicts and win all four of their remaining games they will end on 68 points.
That total would surely push City and Arsenal all the way for the Champions League places but would at the very least guarantee a Europa League spot.