Sam Allardyce's charges have dropped into the relegation zone following a seven-match winless run, with Wednesday's 2-1 loss to fellow strugglers Fulham piling the pressure on the manager.
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The FA Cup has provided little cheer for West Ham in recent seasons and they will be out to avoid a third consecutive defeat at this stage of the competition.
It has been more than three decades since they won the competition for the third and most recent time, while in 2006 they suffered penalty shootout heartbreak in one of the most memorable finals of the modern era - after a 3-3 draw against Liverpool, with Anton Ferdinand's spot-kick miss proving decisive at the Millennium Stadium.
Forest and West Ham met in the fourth round three seasons ago, when a Victor Obinna hat-trick helped the Londoners to a 3-2 triumph.
There have been two league meetings since then, the Hammers completing a Championship double in the 2011-12 campaign on the way to promotion via the play-offs.
The hosts are two-times winners of the FA Cup, though their most recent triumph was in 1959, and the first came in 1898.
Manager Billy Davies's main aim will be securing a return to the top flight after a 15-year absence but progress in the cup brings notable financial reward and a Premier League scalp would underline the club's potential.
Forest are fifth in the second division, unbeaten in seven matches and seven points adrift of the automatic promotion places.
The City Ground has been a tough place for visiting sides this season, with Forest having lost just twice on home soil in 12 matches.