The Vixens currently sit second in the Women's Championship and have lost just one of their past five matches.
That defeat came against City in the quarter-finals of the Continental Cup earlier in January, as Gareth Taylor's side ran out 6-0 winners.
However, Bristol bounced back with a 4-0 win over Oxford United in the FA Cup fourth round to book their place in the last 16.
The draw was conducted by former City goalkeeper Karen Bardsley on the BBC's Sportsday show.
The latest Women's FA Cup fixtures are scheduled to be played on Sunday 26 February. Keep across mancity.com and the official Man City app for further updates.
Gareth Taylor's side earned our place in the fifth round with a convincing 7-0 triumph over Sheffield United.
A Khadija 'Bunny' Shaw hat-trick, Julie Blakstad brace and further strikes from Chloe Kelly and Deyna Castellanos set the seal on a spectacular attacking display at the Academy Stadium.
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City have won the FA Cup three times - most recently in the 2019/20 competition having also lifted the trophy in 2017 and 2019.
Gareth Taylor's side finished as runners up in the most recent edition of the tournament, falling to defeat against Chelsea in extra time at Wembley Stadium.
Charlton Athletic v Birmingham City
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Lewes v Cardiff City
Tottenham Hotspur v Reading
Chelsea v Arsenal
Brighton & Hove Albion v Coventry United
Bristol City v Manchester City