Britain's three Uefa Cup representatives have been handed favourable ties in Friday's draw at Uefa's Nyon headquarters.
Liverpool will face Levski Sofia, with the first leg at Anfield on 26 February and the return on 3 March.
Newcastle, the last team out of the 32-team draw in Switerland face Norwegian side Valerenga.
Celtic, knocked out of the Champions League on Wednesday, have the consolation of a Uefa Cup tie with FK Teplice of the Czech Republic.
After the draw Liverpool chief executive Rick Parry said he hoped the club could emulate their success in 2001, when Gerard Houllier's side lifted the Uefa Cup.
"We have a lot of experience in cup competitions," he said.
"If you think back, 2001 was hugely enjoyable. It would be magnificent if we
could go so far again.
"We haven't been to Bulgaria for a while. It continues our eastern odyssey
this year after Slovenia and Romania."
Third round draw in full:
Brondby v Barcelona
Parma v Genclerbirligi
Benfica v Rosenborg
Marseille v Dnipro
Celtic v FK Teplice
Perugia v PSV
Groclin Dyskobolia v Bordeaux
Valencia v Besiktas
Galatasaray v Villarreal
Club Brugge v Debrecen
Sochaux v Inter Milan
Liverpool v Levski Sofia
Spartak Moscow v Mallorca
Gaziantepspor v Roma
Auxerre v Panathinaikos
Valerenga v Newcastle