The 28-year-old will undergo a medical and finalise the terms of his switch before the transfer window closes at 11pm.
Goal understands that United will pay a £10 million-plus fee for the Colombia international, as well as a significant part of his £300,000-a-week salary.
United are also expected to complete a deal for Ajax defender Daley Blind before the deadline which will take United’s total spending this summer to a figure close to £160m.
Earlier on Monday, Javier Hernandez completed his season-long loan switch to Real Madrid. The Spanish club also have the option to buy the Mexico international.
Manchester City appeared to be closing in on the signing of Falcao after the striker’s proposed move to Real Madrid fell through at the end of last week.
The Colombian, who required surgery to repair a damaged anterior cruciate ligament in January, was targeted by City as a replacement for Alvaro Negredo, who has agreed to join Valencia on loan.
Arsneal manager Arsene Wenger had also sanctioned a pursuit of Falcao with the same agreement as United have reached with Monaco.