Rafael Benitez's side host Swansea at the Stadio San Paolo after a goalless draw in Wales, knowing a score draw would eliminate them on away goals, as well as a defeat.
Tottenham face arguably a greater task in overturning a 1-0 deficit against Dnipro of Ukraine when the two sides meet at White Hart Lane.
Yevhen Konoplyanka's late penalty earned the Ukrainian side a slender advantage in the first leg as the English side's 100 per cent record in the competition was snapped.
Should Tim Sherwood's side progress, they are likely to face last year's runners-up Benfica, who hold a 1-0 lead going into their home leg with PAOK.
Two sides expected to make it through are Juventus and Fiorentina, who both hold two-goal cushions over their opponents.
Juventus beat Trabzonspor 2-0 at home last week, while Fiorentina won 3-1 in Denmark against Esbjerg, and the Italian duo will face each other in the next round if they avoid a shock.
Lazio could not complete a trio of Italian winners however, as they went down to a 1-0 home reverse at the hands of Bulgarian outfit Ludogorets, leaving Edoardo Reja's men with it all to do in the return leg.
Valencia look set to progress, holding a 2-0 lead over Dynamo Kiev with the home leg to come, while AZ will also be confident after a 1-0 win in the Czech Republic against Slovan Liberec.
One team with a mountain to climb are Dutch giants Ajax, who have to somehow turn around a disastrous 3-0 defeat against Red Bull Salzburg in Amsterdam.
Local rivals domestically, Real Betis and Sevilla will look to boost the presence of La Liga teams in the last 16, with both earning score draws from away trips in the first leg.
Betis returned from Rubin Kazan with a 1-1 draw, while Sevilla's game at Maribor ended 2-2. If both sides progress, they would meet in a spicy last 16 tie.
Two sides knocked out of the UEFA Champions League, Shakhtar Donetsk and Viktoria Plzen, go head to head with the tie delicately poised following a 1-1 draw in the Czech Republic in the first leg.
Porto, winners of the tournament in 2011, let a two-goal lead slip at home to Eintracht Frankfurt last week to draw 2-2 and their misery was further compounded as an 81-game unbeaten run at the Estadio do Dragao was brought to an end by Estoril at the weekend.
Coach Paulo Fonseca has seen his future questioned in the aftermath of the defeat, and will be desperate for a win, though Frankfurt's two away goals will give them confidence.
Elsewhere the ties between Anzhi and Genk, Chornomorets and Lyon, and Maccabi Tel Aviv and Basel also remain finely balanced following goalless draws in the respective first legs.
