According to Mundo Deportivo, the Poland international has agreed personal terms on a two-year deal with Madrid after a series of meetings took place this month.
Madrid president Florentino Perez decided to make a move for the Bayern striker as he waits to see how the potential deal to sign Neymar from Paris Saint-Germain evolves.
Real have been chasing Lewandowski for some time and managed to fight off interest from Chelsea, PSG and Arsenal to speak to his representatives first, with the forward understood to be keen on moving to the Bernabeu.
Madrid had also been interested in signing Tottenham striker Harry Kane, but were pessimistic about their chances of securing his signature with chairman Daniel Levy refusing to hold negotiations.
That subsequently turned Perez’s attentions to Lewandowski, who has a proven track record and has scored 32 times for the Bundesliga side this season.
He is believed to be a direct replacement for French striker Karim Benzema, who has endured a disappointing campaign with a modest goal return of just seven goals in 31 games.
Madrid are also keen to bring Chelsea winger Eden Hazard to Spain. The Belguim international is entering the final 15 months of his contract and is yet to sign a new deal.
Hazard is understood to be keeping his options open amid Chelsea’s battle to secure Champions League qualification for next season and could be ready to quit if they fail to make it.
Depending on who Madrid sign in the summer, it could have a knock-on effect on Gareth Bale, who has attracted interest from several Premier League clubs.