The Premier League side will leave the Boleyn Ground at the end of the campaign.
Slaven Bilic's men are gearing up for their final league game at the Boleyn Ground before they make the switch at the end of this campaign. And McAvennie - who was part of the Hammers side that finished third in the First Division back in 1986 - believes that the club now have the potential to move back towards the summit.
"The new stadium will be tremendous and maybe West Ham can challenge for the league," he told the Record.
"They've got to move as they've sold over 50,000 season tickets. People asked whether they would have sold that amount - and they wouldn't have playing the type of football big Sam Allardyce did - but it's different class with Slaven Bilic."
While many would have previously laughed at McAvennie's suggestion, Leicester City's achievements this season prove that practically anyone in the top flight has the potential to go all the way and win the title.
And with West Ham playing much better football than the Foxes often did in the seasons before they pulled off the unthinkable, it does seem possible that another team outside of the established elite - which the Hammers are already helping to break up - could go on to stun the rest of the division.
And with their new facilities potentially making it easier for them to entice better players to the club, it appears that while they are unlikely to be one of the favourites at any stage in the near future, they tick some of the boxes that will help their cause.