But McDermott, who has also enquired about Arsenal teenager Serge Gnabry, has not given up all hope on Becchio after being told by Norwich counterpart Chris Hughton that the Argentinian forward was currently unavailable.
"You'll always have calls for new players. That's how it works," McDermott told the club's official website. "I made an enquiry to Chris Hughton regarding Luciano Becchio and he wasn't available.
"Will it change? Who knows? Things do. I made an enquiry for a couple of young players at Arsenal as well - Gnabry who scored their first goal on Saturday was one of them.
"I wouldn't think they will let him out now.
"It's a tricky situation because Premier League clubs can't just go out and loan anyone - they want better players too if they are strengthening - so they don't want to be left short in their own squads."
Becchio left the club last January after making over 200 appearances for Leeds, while Arsenal striker Gnabry has made seven first-team appearances for Arsenal and scored their opening goal in his side's 2-0 win at Swansea on Saturday.
Becchio is willing to abide by Norwich's decision to deny him a return to Leeds, but is likely to push for a move back to Yorkshire or elsewhere if he does not get any game time before the January transfer window.
Leeds manager Brian McDermott on Monday confirmed he had seen an enquiry for the Argentinean blocked by counterpart Chris Hughton, denying the 29-year-old striker a move Press Association Sport understands he would have welcomed.
It is understood that former Barcelona B man is "frustrated and not very happy" at how his time at Carrow Road is panning out.
It is understood Becchio was keen on a move to Leeds, a club that is "a very nice place" to him, although as things stand Hughton appears unwilling to sanction a move, especially with summer signings Johan Elmander, Gary Hooper and Ricky van Wolfswinkel still bedding in.