The star was edged out of contention by Bruno Mars and Kendrick Lamar, who each received multiple nominations.
Jay Z leads with eight nominations, including his first ever appearance in the album of the year list for 4:44, which comes 21 years after his debut.
Sheeran did receive several nominations in the pop categories, however.
Shape of You is up for best pop solo performance, while ÷ is shortlisted for best pop album.
Lady Gaga is also relegated to the pop categories for her recent album, Joanne, while Katy Perry fails to receive any nominations at all.