The Hammers face Blackburn Rovers in the fifth round.
The Hammers beat Liverpool 2-1 at Upton Park on Tuesday in a fourth round replay, with a late headed goal from Angelo Ogbonna sending them through.
That was their third win against the Reds this season, in what has been an impressive first campaign under Slaven Bilic.
And with Blackburn Rovers now their opponents in the next round, Nolan says that the Hammers should be targeting FA Cup glory as a fitting send off for Upton Park.
“I think this year would be absolutely fantastic if they can cap it off with a trophy before moving,” he told BT Sport.
“With the way the draw has gone, there’s a few Premier League ties and they’ve got Blackburn, so they’ve got to be confident to go there, get the victory and go into the next round.”
“They’ve beaten everyone above them this, so they’ve got to be thinking ‘yeah we can do it this year.’”
Slaven Bilic’s side will certainly fancy their chances of being successful in the competition given the ties that the so-called big sides have been given.
Chelsea are hosting Manchester City, Arsenal have a potentially tricky tie against Hull City to negotiate while Tottenham Hotspur are hosting Crystal Palace.
Those ties increase West Ham's chances of making the last eight and beyond considerably and they should definitely be eyeing a long run in the competition after seeing off Liverpool.