The 29-year-old Colombian striker joined Manchester United on transfer deadline day in September, initially on a temporary basis and with an option to make the move permanent.
Falcao made 29 appearances and scored four goals during nine months at Old Trafford, with his last goal coming against Leicester on 31 january.
The columbian who missed the2014ld Cup after suffering a serious knee injury has previously played for FC Porto and Atletico Madrid.
The Monaco striker made a point of waving to the United fans when he was substituted in the 1-1 draw with Arsenal on 17 May, in what proved to be his last appearance.
And he departs with the best wishes of the manager and everyone at the club.
“Falcao is a top professional and a good human being. On behalf of myself and everyone at the club I would like to wish him well for the future. On behalf of myself and everyone at the club I would like to wish him well for the future.” said Louis van Gaal.
£265,000-a-week wages and exorbitant transfer fee for a 29-year-old ultimately forced United to turn down the chance to sign him permanently.
Falcao will now be looking to find himself a new club as Monaco are facing financial troubles.