The Brazilian has been offered to both Atletico Madrid and Los Blancos but the former have turned down the opportunity.
Jesus is about to enter the final year of his contract at Manchester City so would be an inexpensive transfer.
It is expected he would demand a salary in the region of €5m per season. Still just 25 years old, he could offer an interesting alternative to Karim Benzema through the middle for Madrid.
Jesus, who has 55 caps for the Brazilian national team to his name and 19 goals, contributed 13 goals and 12 assists to City across 41 appearances in all competitions last season.
Pep Guardiola’s side won the Premier League for the fourth time in five years although they suffered disappointment in Europe. Madrid knocked them out of the Champions League.