Chris Waugh of the Daily Mail cited Spanish newspaper AS as claiming that the Colombia international has been in Madrid to finalise the deal.
And Tomas Roncero of AS claimed on Sunday that the start of the transfer process had begun.
At the Vicente Calderon Stadium, Falcao was a totemic figure for the team, scoring 32 goals for Diego Simeone's team.
Atletico found a replacement goalscorer in the shape of Diego Costa following the departure of Falcao, but the South American's move to Ligue 1 came as a surprise.
Falcao had previously been linked with the Bernabeu at the end of the 2012-13 La Liga season, and Paulo Freitas of Sky Sports reported that the forward had denied that the move to Monaco was a stepping stone to Real.
Falcao insisted he was at Monaco for the long-term, and was quoted as telling El Heraldo newspaper at the time:
I don't think like this.
Monaco enticed me because they have a long-term project, because their board are serious and because I love challenges.
But there were no signs that Falcao was easing up on his form last season, as he struck nine goals in 17 league appearances.
However, his hopes of enhancing that tally were ended in January when he suffered anterior cruciate ligament damage during a French Cup tie.
The injury also forced him out of the World Cup finals with Colombia, although the national team have now reached the quarter-finals with the help of Falcao's Monaco team-mate James Rodriguez.
Former Real defender Ricardo Carvalho believes the potential arrival of Falcao would form a fierce forward triumvirate at the Bernabeu alongside Cristiano Ronaldo and Gareth Bale, as Luke Augustus of the Daily Mail reported.
The ex-Chelsea star played alongside Falcao for Monaco last season, and he's quoted as saying:
Everyone knows Cristiano Ronaldo is the best player in the world and also the best professional.
In terms of professionalism, Falcao is the player who is closest. He will have a very long career because he takes so much care.
Cristiano, Gareth Bale and Falcao would be the most feared three-man attack in football history. But it’s better (for Monaco) that Falcao stays.
I think Falcao is the missing piece to make the Madrid attack perfect. Cristiano is the best player in the world, Bale is very good and Falcao is the best striker there is. I’m a defender and I wouldn’t want to face them!
However, it remains to be seen whether Falcao will be linking up with Ronaldo and Bale in such a forward line.