Johnson insists he has not agreed a contract with the Australian Rugby Union and is currently considering Connolly's offer.
Johnson is currently in charge of Wales following Grand Slam winner Mike Ruddock's departure last month.
He also has pressing family issues to contend with in Australia, although Wales remain hopeful he will stay, either as head coach or in his previous role of skills expert.
Connolly is thought to want Johnson and current Bath forwards coach Michael Foley working alongside him as he begins planning Australia's 2007 World Cup strategy.
"I have not agreed a contract with the Australian Rugby Union," said Johnson.
"I have met with their new head coach John Connolly, and I am considering an offer he has made me to join his coaching team.
"But I have said all along that I would love to remain in Wales if I can resolve some of my personal issues in Australia.
"I have kept Steve Lewis (Welsh Rugby Union chief executive) and the team fully informed of my situation and, in effect, nothing has changed this week from the last few months.
"As soon as I know what is possible in my life I will inform Steve Lewis and the players, but I certainly won't be conducting any of my business through the media.
"At present, I am concentrating all my efforts on preparing the team for the games against Italy and France, and I won't be allowing anything to get in the way of that."
Lewis, meanwhile, will present a proposal to the WRU board of directors this month on how the new Welsh coaching team will be recruited.
Lewis has been charged with finalising details on a recruitment plan to assemble the best possible coaching team ahead of Wales' 2007 World Cup campaign.
"As soon as Mike Ruddock informed us that he did not wish to extend his contract as Wales national coach last month, we began to formulate a recruitment plan for our new coaching team," said Lewis.
"We had been negotiating with Mike for over nine months, while we are in a similar position with Scott. He (Johnson) has told us he wants to stay in Wales, but still has some issues to sort out back home in Australia.
"He has kept us fully informed on a regular basis of what those issues are.
"While he is currently concentrating on preparing the team for the Six Nations games against Italy and France, we have agreed that we will meet immediately after the championship to thrash out his contractual situation.
"Scott's current contract ends next month and we would obviously like him to stay - whether that be as head coach or in his former capacity as skills coach.
"Due to Mike's departure, the WRU will have to recruit new blood to the national coaching team, and that is the issue I am currently addressing with the board."