McAvoy signed for Liverpool from his hometown club in December 1947. He was 29 years old at the time and struggled to earn a place in the defending champions' team and made only a single appearance for the club in the rest of the season, that debut coming when Stoke City visited Anfield on 3 January 1948, replacing Cyril Done. The same thing happened the following season too; just a solitary appearance against Chelsea in October 1948. The left-winger played his best years in Scotland with Kilmarnock. In 1939 he was sent along with a group of his compatriots by the Scottish F.A. to tour USA. They were promoted in the U.S. as "The Scottish Internationals" even though only seven of the 17 in the squad had ever played for Scotland. Bill Cole, the "soccer" writer for the Toronto Star blasted the Scottish F.A. for sending a "weak team".