According to the Daily Mail, United boss David Moyes has dispatched his scouts to watch the international friendly between Falcao's Colombia and Belgium in Brussels on Thursday.
Reports earlier today revealed that Chelsea boss Jose Mourinho will make a personal check on the Monaco forward in the Belgian capital, ahead of a potential January bid.
The Stamford Bridge manager is frustrated with the lack of goals from his current first-team options, and Falcao - who is also eligible for the Champions League latter stages - has now emerged as one of his top targets for the winter window.
Having just moved to Monaco in the summer, the report claims Chelsea have earmarked up to £60m to put towards a Falcao transfer.
But the Blues could face competition from United - although the defending Premier League champions are unlikely to submit a bid of their own in January.
Continuing uncertainty over the futures of Wayne Rooney and Javier Hernandez means that United are still monitoring potential striker targets with a view to reinforcements next summer if necessary.
But while United's need is not urgent, Chelsea's attacking struggles mean Mourinho's requirements are more pressing. The Chelsea boss missed out on the likes of Rooney over the summer, and only brought in Samuel Eto'o from Anzhi.
As well as United, Chelsea could face competition from Real Madrid - long-term admirers of the Colombian since his days at cross-town rival Atletico.