Following a two over par round of 73 on Thursday's opening day, Aberg went racing up the scoreboard on day two, when he carded a four under par round of 67, moving to two under for the championship.
But the 2024 Masters runner-up was still eight shots behind Scheffler at the midway point, tied for 17th place.
In today's third round, Aberg made an identical start to yesterday, with birdies on the second and third holes. But though the 25-year-old survived all 18 holes without a single bogey, he failed to cut into the advantage of the leaders, registering one more birdie and scoring 25 pars on his round.
A score of -3 for the round leaves him on -5 and tied for 14th for The Open with one round to play, but he is further behind Scheffler than he was at the start of the day, as the American carded a round of 67 to move onto -14.
A first major win for the world number nine is by no means impossible, but a collapse from Scheffler is one of many things that must fall his was on the final day in County Antrim.
World number 135 Jesper Svensson matched his compatriot's round of 68 - his best of the championship so far - to move onto -2 and is now tied for 34th. Henrik Stenson needed one shot more to complete today's round, and is now at -1 and tied 44th.
However, one Swede who had a bad day was Sebastian Soderberg, who shot a four over round of 75 and now sits tied for 66th in the field reduced to 70 competitors following yesterday's cut.