McKay, currently chief operating officer with the Scottish Rugby Union, will succeed Peter Lawwell in the hot-seat at Parkhead when the latter retires on July 1.
SRU chief executive Dodson has played down the prospect of McKay joining Celtic sooner as the club steps up its search for a successor to Neil Lennon whose resignation as manager was announced on Wednesday.
An increasingly febrile atmosphere has surrounded the Glasgow club as their bid for a tenth successive league title has unravelled but Dodson believes McKay is equipped to handle the pressure.
“He’s been around the block,” the rugby chief told journalists. “He’s seen me suffer at the hands of you guys and others. He’s seen that you can get credit for things that you haven’t had a lot to do with, and that you get criticised for things that you’ve not been responsible for.
“If you take a job as a CEO you have to expect that you develop a tough skin, and he’ll probably need one wherever he goes in business. He knows what to expect.”
Lawwell, Lennon and Celtic’s major shareholder Dermot Desmond have all found themselves a target for supporters’ ire but Dodson thinks McKay has the skill-set to handle the switch to football.
“He has been here a decent period of time and he has been instrumental in the transformation of the organisation from a commercial and sporting perspective. He is a really talented guy. I'm close to him and I know what a good operator he is.
“There are different pressures in the two sports. But he is a smart individual and a very experienced operator and I’m sure he will adapt.”
Dodson will now cast his net wide to find a successor for McKay but stressed that the chief operating officer still had an important role to play in his remaining time with Scottish Rugby.
“From my point of view, what's happening at Celtic is something that is out of my control,” said Dodson. “I clearly know that Dom will have a very full in-tray when he gets there. The issue for us is that he has a notice period here and we need stability in our business too. I expect him to be here with us for a while yet.
“So until we’ve stabilised our business, until we can replace and make sure the business is in good enough shape, that’s the only time we’ll consider allowing Dom to move across to Celtic.”
Asked if an offer of compensation from Celtic would allow them to secure McKay sooner than July 1, Dodson replied: “It’s not about the money. That’s a short-term fix. If you get into those kind of compensation deals, then the most important thinking is making sure this place can run and function properly. That’s my job as CEO, to make sure, selfishly, that we look after this business.
“If there is a point in time when we can let Dom go earlier than that then I’m sure we’d try to work around that, but as it stands at the moment we are some way away from that.”