According to the Sun, Liverpool have contacted Monaco to enquire about the on-loan United striker.
However, Chelsea are also now thought to be keen to sign the Colombian.
Brendan Rodgers' side want to sign Falcao on loan, similar to his current deal at Old Trafford.
However, Chelsea boss Jose Mourinho is keen to use his relationship with the 29-year-old's agent Jorge Mendes, to thrash out a deal and beat the Reds to his signature.
Falcao has struggled to settle into life in the Premier League since his move from Monaco, having scored just four goals in 22 appearances for the Red Devils.
GETTY
Liverpool and Chelsea are keen to sign Radamel Falcao on loan
He played just seven minutes during United's 2-1 win over Liverpool on Sunday, replacing Ander Herrera late in the second half.
Falcao's friend and former agent Silvano Espindola has recently blasted Louis van Gaal for reducing the Colombia international to tears after making him play for the under-21s.
Espindola said: “When he was on his way to the ground to play this game with the reserves, he called me.
“We spoke for 20 minutes until he reached the stadium and he told me, ‘I’ve never been through something like this, I don’t know how to deal with this situation. It feels weird for me!’"
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