Celtic claim a fee was agreed with Manchester City for John Guidetti prior to them taking the striker on loan.
Xabi Alonso claims no Premier League offers were put to him before leaving Real Madrid for Bayern Munich.
Beckford has scored three times in eight appearances since arriving at the John Smith's Stadium at the end of September.
And the latest agreement keeps the striker with Town until 31 May.
"This decision came naturally," said Beckford, whose opportunities have been limited by hamstring trouble. "Simon Grayson and the players have shown a lot of faith in me. It feels like the right place to be right now.
"The reaction of the fans always plays a part in this sort of decision too, because they are the fabled 12th man. For the Huddersfield Town fans to take me in like they have done is a contributing factor to me wanting to extend the loan."
Boss Grayson told the club website: "I think people have seen his qualities and what he can bring to this football club when he's been in the team. He has a real hunger and desire to succeed here and do well for us.
"Unfortunately we've missed Jermaine a bit through injury, but hopefully he will get a clear run now and get plenty of games and goals under his belt."