Having allowed Wayne Rooney to sign for Derby County, the MLS side are searching for another marquee name to replace the England legend.
According to the Daily Mail, Modric is on their radar, with the Washington-based outfit looking to secure a deal for the Croatian in the coming months.
Despite 18 months still remaining on his contract, Real are said to be open to offers for the 34-year-old as they bid to reduce their wage bill.
With the veteran allegedly open to a new challenge, discussions have been held between the American side and Modric's representatives.
One potential stumbling block regarding an imminent move is Modric's likely unavailability during the summer due to representing Croatia at Euro 2020.