Pompey lost Awford's first match in charge against Southend United on Tuesday night, but he says that he would "consider" becoming manager if the club's board approached him.
"I haven't applied and I haven't ruled myself out," he told The News.
"The club know where I am, they know me and what I'm about. They know what I do and what I don't do.
"So if they would like to talk to me about the job, I'm sure they will and then I would consider it."
Pompey are 18th in League Two and face Hartlepool United on Saturday.