The 26-year-old from Dunfermline is out of contract this summer, meaning she is available on a free transfer.
A final agreement has not yet been reached but The Daily Telegraph is reporting that the Spanish giants are “increasingly confident” of securing Weir's signature.
Weir has been linked with Real Madrid in the Spanish media for several weeks and has also been attracting interest from elsewhere around Europe, Madrid appears to be her favoured destination.
It would complete a monumental and inspirational rise to the top for a player who started her career at Hibs as a ten-year-old.
She made her way through the Hibs Girls youth system before making her first team debut in May 2011 while still at school.
Weir joined Arsenal in July 2013 after leaving high school and had spells at Bristol Academy and Liverpool before joining City four years ago. She’s since become a key player in Women’s Super League.
The Scotland international with the best left foot in the women’s game has been a nominee for Fifa's Puskas award, for the world's best goals of the year, for the past two years.
Now regarded as one of the most technically gifted female players in the world, Weir was one of two Scots selected by Great Britain for the 2020 Summer Olympics and has won 87 Scotland caps, scoring 13 times for her country.
Madrid finished third in the Spanish league this season, with Barcelona winning the title. A Women’s Champions League match between the two rivals at the Camp Nou in March attracted a world record 91,533 crowd.