He was the fifth of nine children of Richard and Anne Avison who lived in the house beside St. Bartholomew's Nunnery in Nolt Market, Newcastle. They were both were musicians and were presumably Avison's first music teachers.

Richard was a member of the ancient Incorporated Company of Town Waits i.e. a member of the official town band, who was licensed to teach music in his spare time. He was paid the very small salary of £4 per annum (plus a uniform - valued at £5 when a cash substitute was offered) which he presumably supplemented with teaching. His wife Ann was a church organist.

As Avison's first published composition (Six Sonatas for Two Violins and Continuo) was dedicated to Ralph Jenison (1696-1758) it is presumed that Jenison had some early connections with the composer. Jenison, a patron of the arts, was from an old Newcastle family and was elected Member of Parliament for Northumberland from 1734 to 1741.

Jenison's association with music is evidenced by the fact that Johann Pepusch (1667-1752) dedicated his edition of Corelli's concertos to him. Avison was later assisted in his studies by Colonel John Blaithwaite who was a retired director of the Royal Academy of Music.

During the period with Jenison, Avison must have moved to London, as according to the music historian Charles Burney, he studied music there with the Italian composer Francesco Geminiani (1687-1762).