Early reports had suggested the dog, called Colonel, was from the US but sources in Washington have denied this.
The dog was taken during an operation on December 23 last year east of Kabul.
Captain Richard Holloway, 29, was killed during the mission. He was from the Royal Engineers Regiment but served with the SAS.
Capt Holloway has been described by his commanding officer as a "natural leader", whose "tactical ability commanded wide respect".
He added: "His judgement was un-erring, his enthusiasm was infectious and his standards never dropped."
The military regularly use dogs for surveillance and bomb detection.
The fate of Colonel is not known. The Ministry of Defence has not commented on the reports.
But a statement from the International Security Assistance Force (ISAF) said: "We can confirm that a military working dog went missing following an ISAF mission in December, 2013.
"It is ISAF policy to defer identification to the appropriate national authorities."