Jurgen Klopp's side advanced from the last-16 after beating Bayern Munich 3-1 at the Allianz Arena on Wednesday night.
The draw was held at UEFA's headquarters in Nyon this morning and the Reds were drawn against a familiar opponent from last year's Champions League run.
Liverpool play FC Porto at Anfield in the first leg that takes place on Tuesday 9 April.
The second leg is at the Estadio do Dragao on Wednesday 17 April. Both matches will be covered by BT Sport, with kick off set for 8pm (BST).
The Reds played Porto in the Champions League last year, winning 5-0 on aggregate during their last-16 tie.
Should the Reds overcome the test of Sérgio Conceição's side once again, they face either Barcelona or Man United in the semi-finals, with the first leg to be played away on April 30/May 1 and the second leg at home on May 7/8.
Full Quarter-Final Draw
‣ Ajax vs. Juventus
‣ Liverpool vs Porto
‣ Tottenham vs. Man City
‣ Barcelona vs. Man United
Full Semi-Final Draw
‣ Tottenham/Man City vs. Ajax/Juventus
‣ Barcelona/Man United vs. Liverpool/Porto