Paul Hurst’s side face the Hammers on Sunday, January 7 live on one of the BBC channels with a 2pm kick-off.
The figure is reported to be worth £144,000 to each televised club, while winners earn £67,500.
The game will bring former player David Moyes back to Shrewsbury, where he played between 1987-1990.
While it also caught the headlines because of former Town goalkeeper Joe Hart, who is on-loan at the London Stadium from Manchester City.
Hart, from Shrewsbury, was a graduate of the club’s academy and sold to City in 2007.
“It’s a good draw against a Premiership team”, said Hurst.
“There’s part of you that would like to be away from home to get the opportunity for the players to play at the stadium and for the fans to travel somewhere different, but at the same time if you get a home tie it gives you that little bit more chance of causing an upset.
“They’ve had some issues and a change of manager but they’ve got some very good players and we know that it will be a very tough game when it comes around.”