The Colombia striker, 28, has scored one goal in eight appearances sincesigning for the season from Monaco.
Earlier this month United boss Louis van Gaal said Falcao was only fit enough to play for 20 minutes, but Falcao says he is now "almost 100%".
"I am very happy in this city, in this club. I really want to stay," he said.
United have an option to sign Falcao, who is earning approximately £265,000-a-week in Manchester, for £43.5m at the end of the season.
But the striker, who missed the World Cup due to knee ligament damage, has had limited impact during his £6m loan deal so far, due partly to injury.