Australian Johnson took over from Andy Robinson and guided the Dark Blues through their recent RBS 6 Nations campaign. Victories over Italy and Ireland and a joint-third place finish was a reasonable return from a squad that ended the previous championship with five defeats and the Wooden Spoon.
Dodson told Press Association Sport: "We are still talking to Scott and have been doing so pretty much since the Six Nations ended. We said we would sit down at the end of the tournament and talk about the next steps and we are still doing that."
But 50-year-old Johnson - who was assisted by former England forward Dean Ryan during the tournament - has refused to confirm whether he will stay on as Robinson's permanent successor until after talks with the SRU.
Press Association Sport understands Murrayfield chiefs are keen to retain the former interim Wales head coach's services but Dodson admits an agreement is still to be reached.
A report in the Daily Telegraph suggested a sticking point on a deal concerns whether Johnson's role will be expanded to cover player development, with the title of Director of Rugby being considered.
But Dodson added: "The Telegraph story is just supposition. I wouldn't like to be too categorical about anything because Scott and I still have a lot to talk about.
"The Six Nations are only just finished and this is going to take a certain amount of time to wash through. But we are very happy with what Scott and Dean did over the whole of the Six Nations period. The talks will take place and we will make an announcement in due course."
Dodson refused to rule out handing Johnson an expanded portfolio but did say that the entire SRU coaching structure could be redrawn. He recently named former Scotland hooker Stevie Scott as interim Edinburgh head coach following Michael Bradley's failed reign but will make an announcement of a full-time replacement following his talks with Johnson.
Dodson said: "We are looking at the whole coaching structure in Scottish rugby. We have several vacancies to fill so what I want to do is take my time and make sure we bring in the right blend of coaches."
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- Stevie Scott