Connolly has offered compatriot Scott Johnson, the man who took over from Mike Ruddock following his shock resignation, a place on his coaching team and held fresh talks with him.
Johnson, whose contract with Wales expires next month, has voiced his desire to stay where he is with negotiations over his future ongoing, but Connolly has not given up hope.
"Scott is mulling over which way to go," he told BBC Radio 5 Live's Sportsweek programme. "We have had a couple of good chats.
"I was in England last Monday and we had another chat. They went very well, thank you.
"But that's one of those decisions Scott will have to announce when he's made his decision. Me pre-emptying anything is unfair to him and it could be unsettling to Wales.
"He's still to make a decision but I would think in the next week or so, we will know where we stand.
"We have indicated our keenness for what type of coaching team we would like to put together.
"It's always difficult when your country calls. That's a fairly strong tug on the heart strings."