Town chief executive Nigel Clibbens revealed a season-long loan deal for Beckford had been agreed between the two clubs, but it collapsed shortly before the transfer deadline on Friday night.
Huddersfield are keen to strengthen their strike-force after the £8 million sale of star man Jordan Rhodes to Blackburn, and they are considering an attempt to revive a deal for Beckford when loan signings are permitted again at the weekend.
"We spent a lot of last Thursday and all of Friday negotiating with Leicester and we moved close to a deal, agreeing terms for a season-long loan," Clibbens told the Leicester Mercury.
"Jermaine travelled to Huddersfield to sign the paperwork and once it got past 10pm we started filling it in. We signed, Jermaine signed and then Leicester decided they didn't want to do the deal, so we didn't get it completed.
"That's part-and-parcel of football and business, and it can be frustrating. Maybe it will come back in the loan window, I don't know at this stage."
Huddersfield boss Simon Grayson, who worked with Beckford during his time in charge at Leeds, added: "We got very close, we almost had it over the line. He wanted to come but we're in the hands of the other football club. Jermaine is a player I know well and I like."