Jovic, 21, scored 10 goals for the German side in their run to the Europa League semi-finals where they were knocked out by Chelsea on penalties and has netted 17 times in the Bundesliga.
Spanish newspaper AS said Madrid would officially announce the signing next week after the Bundesliga season ends.
Jovic, who has made 13 appearances for Serbia, is expected to be the first of several high-profile players to join Real Madrid as the deposed European champions look to respond to a disappointing campaign on all fronts.
(Reporting by Richard Martin, editing by Ed Osmond)
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