Bradford's run to the quarterfinals has been the feel-good story of this season's competition, with the team having already eliminated Premier League sides Chelsea and Sunderland.
Both teams hit the goal frame in a fractious match at Valley Parade, with second-tier Reading almost snatching victory in the closing minutes when Oliver Norwood's free kick eluded everyone in the box and struck the post.
Bradford is looking to reach the semifinals for the first time since winning the trophy in 1911. Reading hasn't qualified for the last four since 1927.
Aston Villa hosts West Bromwich Albion in another quarterfinal match later Sunday.