The Scotland job is currently vacant after Alex McLeish was sacked last month following disappointing displays in European qualifiers against San Marino and Kazakhstan.
McLeish’s side lost 3-0 to the Kazakhs in an embarrassing display, with a slightly better performance being enough to beat San Marino, a side made up of mostly part-time players, 2-0.
Dyer, who has been Clarke’s assistant since 2017, told the BBC: "It's a pinnacle of anyone's career.
"If the country comes for him it would be a great honour. I know he knows that.
"He has done a great job here. He's always left it open that he will decide at the end of the season, that's no secret. At the end of the day, it is a hard one to turn down.
"We will just leave it up to the gaffer. What will be, will be."
Clarke took over as Kilmarnock manager in October 2017, with the club at the bottom of the Scottish Premiership.
By the end of that season, the club were in fifth place, and had set a club record of 59 points.
The Scottish FA is considering two other candidates for the role of Scotland boss, with Aberdeen manager Derek McInnes and Scotland Under 21 coach Scot Gemmill also in the running.
With the squad for Scotland’s next two qualifiers against Cyprus and Belgium set to be announced in the coming weeks, there is a rush for the new manager to be appointed.
Scotland have not qualified for a major international tournament since 1998, and fans will hope the new coach will be able to put that right.