Speaking on BBC Radio Scotland, Professor Jason Leitch said trials of using the two vaccines together have suggested it is safe and effective.
The UK-wide Joint Committee on Vaccination and Immunisation (JCVI) on Wednesday set out interim guidance for a Covid vaccine booster programme.
Speaking about third doses, Professor Leitch said they would be “beneficial” in prolonging protection of patients, particularly those in the more vulnerable age and health groups.
“So people who are going for their flu, like my mum and dad maybe who would go to the GP practice, may also get the Covid vaccine, on that same day or very close to that,” he said.
“And then as we got a bit younger, maybe we would use pharmacies, we’d maybe use mass vaccination centres, we’d maybe use mobile vaccination units like you’ve seen us do over the last few months.
“We’ll plan that between now and the start of that programme.”
The seasonal flu jab was offered to everyone over 55 in Scotland last year, extended to those over 50 where supplies allowed.
Professor Leitch added: "It looks as though not only will we need them [boosters], but it will be a desirable thing to have, especially for those over 50 and everyone who is more vulnerable to Covid who is over 16.”
If the booster rollout goes ahead, it will see a third dose of vaccine offered as soon as possible from September to all over 70s, frontline health and social care workers, residents of care homes for older people, and all over 16s who are clinically vulnerable.
A third dose would then be offered to all over 50s in a second stage.
A final decision will be made before September. This will take into account the Covid situation, additional data, evidence around how long vaccines last for and new variants.
It comes as a new record for case numbers was set on Wednesday, with 3,887 reported.