The draw was held in London with former Chelsea player, Ruud Gullit tasked with pulling out the home teams while ex-England global Paul Ince drew the away sides.
The fixture against will be Unai Emery's first ever outing in the domestic cup competition and he will be relishing the chance to help Arsenal in their quest for silverware.
United will tackle Barnet of the National League at Bramall Lane, while Wednesday will host Luton Town.
Reading also played Sir Alex Ferguson's United side at Old Trafford in a fifth-round clash in February 2013.
Arsenal could play minor league side Solihull Moors, while London neighbours Tottenham Hotspur may face fellow minnows Southport.
The third round is scheduled to take place in the New Year over the weekend of January 5/6, with each of the 32 victors bagging £135,500 in prize money.