Linfield dominated their first-round Champions League qualifier on Tuesday night but were held to a goalless draw by B36 Torshavn.
Peter Thompson saw a penalty saved and Mark McAllister had a goal disallowed against the side from the Faroe Islands but the Blues could not find a goal to take into the away fixture.
The second leg will take place on July 10 with the winners facing Cypriot side AEK Limassol.
Portadown did well to hold Macedonia's Shkendija to a goalless draw in the first leg of their Europa League first qualifying round tie.
David Miskelly pulled off a number of saves as the home side had 10 corners to their opponents' two in the 40-degree heat at the Milano Arena.
Ronnie McFall's Northern Irish side managed to put up a good defensive display to hold Shkendija.
The winners of the July 10 second leg at Shamrock Park will take on Slaven Koprivnica in the second qualifying round.