The Welsh outfit had agreed a deal to sign the Harrogate star as a replacement for the injured Paul Mullin.
According to the Daily Mail, the Red Dragons had a £500,000 bid accepted by their League Two rivals for the 27-year-old.
Armstrong, who has scored once in five games in all competitions this season, had gone on strike in his quest to secure a move away from Wetherby Road.
Harrogate had announced that the centre-forward had joined Phil Parkinson's side on a permanent deal.
But he will now be forced into staying until at least January after the EFL voided his switch to the Racecourse.
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That is because Wrexham failed to submit the paperwork on time before the English transfer window closed at 11pm on Friday night.
An EFL spokesperson told the Mail: "The registration of Luke Armstrong from Harrogate to Wrexham has been rejected on the basis of the paperwork not being submitted by the necessary deadline."
Wrexham meanwhile broke the disappointing news to their fans just five minutes before their clash away at Tranmere in their early Saturday kick-off.
A club statement read: "Wrexham AFC can confirm terms were agreed to sign Harrogate Town striker Luke Armstrong on Transfer Deadline Day, with paperwork submitted for his signing.
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"Unfortunately, the required documentation and clearance requests could not be completed in time and his registration has been rejected.
"As a result, Luke remains a Harrogate Town player.
"The club would like to thank the player for choosing to join Wrexham earlier in the day and are disappointed that this cannot now happen (pending any appeal).
"No further comment will be made at this time."
Wrexham, who won 1-0 against Tranmere, did manage to bring in two players on deadline day.
Arsenal goalkeeper Arthur Okonkwo was brought in on loan following Ben Foster's retirement, while George Evans has joined from Millwall on a two-year contract.