Villa have been drawn against Club Brugge in the last 16, and would face the Premier League leaders in the next round should both teams make it through.
Arsenal, meanwhile, play PSV, with the winners facing either reigning champions Real Madrid or city rivals Atletico Madrid in a last 16 tie that is a repeat of the finals of 2014 and 2016, both of which were won by Los Blancos.
Bayern Munich will take on German rivals Bayer Leverkusen in the first round of the knockouts and Inter Milan take on Feyenoord, who are fresh from knocking out AC Milan in the play-offs.
Barcelona will face Benfica, who they defeated 5-4 away in Lisbon in the league phase last month. Last season's beaten finalists Borussia Dortmund will take on French side Lille.
The first leg of Liverpool's tie will take place at the Parc des Princes in Paris on March 4 or 5, with the return at Anfield a week later.
The clubs last met in the Champions League in the group stage in 2018/19, when PSG won 2-1 at home after Liverpool triumphed 3-2 in England.
Last 16
Quarter-finals
PSG or Liverpool v Club Brugge or Aston Villa
PSV or Arsenal v Real Madrid or Atletico Madrid
Benfica or Barcelona v Borussia Dortmund or Lille
Bayern Munich or Leverkusen v Feyenoord or Inter Milan
Semi-finals
PSV or Arsenal or Real Madrid or Atletico Madrid v PSG or Liverpool or Club Brugge or Aston Villa
Benfica or Barcelona or Borussia Dortmund or Lille v Bayern Munich or Leverkusen v Feyenoord or Inter Milan