The 21-year-old was sold to Elche last summer and scored seven goals with five assists in 34 games.
His performances have attracted interest from other clubs after just one season with Elche.
Celtic face competition from Bournemouth, Ajax, Bologna, and Feyenoord.
TuttoMercatoWeb journalist Pietro Celeste wrote on X: "Bologna is evaluating Alvaro Rodriguez, a left-footed striker born in 2004. In the last season, he was a standout with Elche with 7 goals and 5 assists (193 centimeters). Competition from Celtic, Bournemouth, Ajax and Feyenoord."
The Spanish-born forward is a Uruguayan under-20 international.
He had a loan spell at Getafe, scoring three goals, before leaving Madrid permanently.
He also found the net regularly for Real Madrid Castilla in the Spanish lower leagues and made a few first-team appearances.