Newcastle's youngsters, along with their Hammers counterparts, were one of 32 sides to join the prestigious youth competition at the third round stage.
The cup tie will be the Magpies' tenth successive away draw in round three of the tournament, with a date, venue and kick-off time to be confirmed in due course. The match must take place and be completed before Sunday, 11th December.
For the 2021/22 tournament, players must have been born on or after 1 September, 2003 and before 1 September, 2006 to be eligible to feature in the FA Youth Cup.
The young Magpies have won the competition twice in their history after tasting success in 1962 before a side including Geordie midfielder Paul Gascoigne, lifted the cup in 1985.
Newcastle enjoyed an impressive run to the quarter-finals of last year's competition, beating the likes of Huddersfield Town, Leeds United and Watford before exiting at the last-eight stage against eventual winners Aston Villa at St. James' Park.