The Colombia striker's season-long loan deal will cost United about $9 million and is subject to a medical examination and personal terms being agreed to, the person told The Associated Press. The person spoke on condition of anonymity because the deal has not yet been completed.
Premier League clubs have until 2200 GMT Monday to sign players before the transfer window closes.
It was unclear if the deal would contain the option for Falcao to make his move to the record 20-time English champions permanent at the end of the loan.
Bringing one of the hottest properties in world soccer to Old Trafford would be a coup for United's owners, with the Glazer family under pressure to spend after a rapid decline since Alex Ferguson retired as manager in May 2013 after more than 26 years in charge.
United's need for reinforcements has been highlighted by the failure to win any of its opening four matches under manager Louis van Gaal, who had been hired to halt the slump under David Moyes last season. United's latest setback was a 0-0 draw at Burnley on Saturday, following a humiliating League Cup loss to third-tier club MK Dons.
United, which is listed on the New York Stock Exchange, has already committed to spending about $240 million since finishing seventh last season, with Argentina midfielder Angel Di Maria joining last week for a British-record fee of 59.7 million pounds (then $99 million).
In Van Gaal's 3-5-2 system, Falcao could partner Robin van Persie up front, with captain Wayne Rooney dropping into midfield and assuming the No. 10 role. Those tactics could put Juan Mata's starting place at risk after he arrived in January for 37.1 million pounds (then $61 million).
Falcao joined Monaco from Atletico Madrid last year in a deal worth 60 million euros (then $78 million) and would be leaving a club in the Champions League for one not participating in any European competitions this season.
Monaco owner Dmitry Rybolovlev, a Russian billionaire, spent heavily to establish the team as a footballing power after winning the second division title in 2013, with Falcao the marquee recruit after scoring 52 league goals in two seasons at Atletico. But Falcao's season - and World Cup hopes - were ended following left knee surgery in January and Monaco was beaten to the title by Paris Saint-Germain.
Colombia striker James Rodriguez, who was the top scorer at the World Cup, has already been sold to Real Madrid after just a year on the French Riviera. Losing Falcao is a further blow to Monaco's ambitions and leaves the club with veteran forward Dimitar Berbatov as the club's best option in attack.
The signings of Falcao and Rodriguez were meant to herald a new era for Monaco, one where they could challenge big-spending PSG. But losing such big names deals a significant blow to the club's prospects of competing at the highest level.
Falcao would be leaving a club that attracted less than 10,000 fans at its weekend game for a club which usually sells out the 76,000-capacity Old Trafford.