Odds were slashed from 40/1 to 2/1 late Wednesday evening, and on Thursday even more speculation grew when it was reported that the FAW are to ask Sheffield United permission to speak to the 41-year-old.
The FAW have already interviewed four other candidates including Brian Flynn, Lars Laderback, Chris Coleman and John Hartson, however, they have highlighted Speed as their number one choice.
The rumours are very untimely for Speed, who has been preparing his players for a mouth-watering encounter at Oakwell tomorrow (Saturday) afternoon, and Chief Executive Trevor Birch has labelled the Welsh FA's intentions as 'unhelpful'.
He said last (Thursday) night: "The speculation is very unhelpful as Gary has only one thing on his mind at the moment and that is to prepare the team for the derby game against Barnsley at the weekend.
"Although unwelcome, the Welsh FA's interest is quite understandable as he's not only a Welshman but also a manager with a great future ahead of him. Gary has not been approached and there are no discussions planned.
"He has already stated in today's press conference that he is happy here and we are more than happy with him. This has become an intrusion into his and the club's preparations and we will not be commenting further."
Speed rubbished rumours in a press conference earlier on Thursday afternoon. He said: "It is news to me and I won't comment on speculation based on bookmakers' odds.
"Of course, I am a proud Welshmen and many in my position would jump at the chance to manage their country's team. If it happens at some stage in the future, fine, but I am happy here and concentrating on our game at Barnsley on Saturday."
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