The Tigers reached the fourth round of the FA Cup with a 3-1 victory over Ipswich and Barmby confirmed after the match that he is still on track to take the full-time job.
"Hopefully everything will be sorted by Monday," he told reporters when asked when confirmation is expected that he will retain his role at the KC Stadium.
Barmby was pleased to see some of his fringe men impress as he made seven changes to the side that lost 1-0 at home to Derby last Monday.
This win has now given the former Tottenham man a welcome headache as Hull look to continue their promotion push in the Championship.
"The squad is strong now and you need everyone on their toes," he added.
"I want people coming in and staking a claim because you don't want anyone who thinks they are going to play week-in week-out.
"You really want the competition for places and the lads know that.
"All in all it was a good day for us."
Barmby is looking forward to Sunday's draw for the fourth round but he isn't too worried about who his side will come up against in the last 32.
"It would be great if we got one of the big boys but whoever we get we will treat it as if we are playing one of the big boys.
"It would be nice but we will have to wait and see."