Liverpool will play at Czech side Sparta Prague in the first leg of the Europa League round-of-16 on March 7, just three days before a huge Premier League match against Manchester City.
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Juergen Klopp's league leaders learnt of their opponents in Friday's draw made at UEFA headquarters in Nyon.
Liverpool and the other seven teams that topped their groups will host the second-leg matches a week after their openers away from home.
Avoiding a long round-trip journey to Azerbaijan to face Qarabag in midweek will be a relief for Liverpool, who host City on Sunday, March 10.
"The first thing to say is that at this stage of European competition it makes sense to expect a proper test and this is what we have got," manager Juergen Klopp told Liverpool's website.
Instead, German league leaders Bayer Leverkusen have again been paired with Qarabag after they also met in the group stages. Leverkusen won 5-1 at home and 1-0 in Baku.
Leverkusen coach Xabi Alonso is one of Europe's most in-demand strategists with upcoming vacancies at Bayern Munich and Liverpool.
West Ham beat Freiburg home and away in the group stage, and the two teams will also meet again in the last 16.
Roma advanced on Thursday with a penalty shootout victory over Feyenoord in their play-off, and next face Premier League Brighton.
In the third-tier Conference league, Ajax were drawn to play EPL giants Aston Villa.
The English club topped their group to qualify directly for the round of 16.
Ajax advanced on Thursday when Kenneth Taylor scored the winner in extra time to defeat Norwegian host Bodo/Glimt 2-1 and 4-3 on aggregate in their knockout playoff.
The Europa Conference League final will be played on May 29 in Athens, with Greek clubs PAOK and Olympiacos still in the running.
EUROPA LEAGUE ROUND-OF-16 DRAW
(Group winners listed first, playing away on March 7):
Atalanta v Sporting
Bayer Leverkusen v Qarabag
Brighton v Roma
Liverpool v Sparta Prague
Rangers v Benfica
Slavia Prague v AC Milan
Villarreal v Marseille
West Ham v Freiburg
CONFERENCE LEAGUE ROUND-OF-16 DRAW
(Group winners first, playing away on March 7):
Aston Villa v Ajax
Club Brugge v Molde
Fenerbahce v Union Saint-Gilloise
Fiorentina v Maccabi Haifa
Lille v Sturm Graz
Maccabi Tel Aviv v Olympiakos
PAOK v Dinamo Zagreb
Viktoria Plzen v Servette
Australian Associated Press
