The 29-year-old will return to parent club Monaco, at least initially, after four goals in 29 appearances was not deemed good enough for United to consider splashing out the reported 55 million euros ($60.5 million) fee to buy him.
"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," manager Louis van Gaal told ManUtd.com after the 0-0 Premier League draw at Hull City left United fourth in the table and in the Champions League playoff spot.
Falcao was once one of the hottest properties in world football but a curious move to Monaco and a serious knee injury have curtailed his career.
United have already recruited forward Memphis Depay from PSV Eindhoven and with Van Gaal settling on one up front during the team's best run in the third quarter of the campaign, another frontman may not be needed.
Borussia Dortmund defender and Germany World Cup winner Mats Hummels has been repeatedly linked with United in media reports.
(Writing by Mark Meadows, editing by Ed Osmond)
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