Strachan left today after failing to guide Scotland to the World Cup.
Scotland needed to beat Slovenia on Sunday for a shot at the play-offs but could only manage a 2-2 draw.
And Moyes, who left Sunderland in the summer, is the bookmakers’ favourite for the gig.
Alex McLeish has already thrown his hat into the ring and is fourth favourite behind Malky Mackay and Derek McInnes.
Full list of odds from Ladbrokes
25/1 B Davies
25/1 W Smith
20/1 S Clarke
25/1 H Redknapp
33/1 G Souness
100/1 A Ferguson
Strachan said: “I said on my first day as Scotland manager that it was the proudest moment of my career and that I wanted to put a smile back on the nation’s face again. I share the profound disappointment at missing out on the play-offs, especially having worked so hard to fight our way back into contention.
“The players should receive immense credit for that resilience in coming back from a difficult start and I would like to thank each and every player who has come in to represent their country.
“Together we have shared some really magical moments and those memories will live with me for ever.
“Of course, what made those moments special was the unwavering support of our fans. Through highs and lows you have stayed with the team and my biggest regret is not being able to provide the finals tournament you deserve.
“I would like to thank the Board, the backroom staff and everyone at the Scottish FA for the help and support they have given me as national coach. It has been a real privilege.”