He said the move would not be "cheap" but argued it could bring economic and social benefits to rural communities.
Mr Stewart made the comments following a "fact-finding" visit to the Faroes, a series of 18 islets in the North Atlantic, last week.
He said: "I've got an open mind – it may not be practical.
"But given the population sizes are broadly comparable, it seems worth exploring.
"Particularly when you look at the wider cost-benefit analysis in terms of social issues, stopping depopulation, improving healthcare, decarbonisation."
The Scottish Government has proposed exploring the construction of fixed links in the Western Isles and between Mull and the mainland.
Mr Stewart said the tunnels in the Faroe Islands cost about £20 million per kilometre.
He said: "It's not going to be cheap, but you contrast that against building several new ferries – not just once, but given those tunnels would be there for many, many years, probably two or three cycles of building ferries."
He suggested funding could come from the UK and Scottish governments and local authorities, as well as through possible tolls.
"If these tunnels are built, you are looking at many generations of sustainability," he added.
He added: "If this was to be progressed, the next stage would be a feasibility study where you would assess the complexities of rock structure and everything else."
Mr Stewart said local communities would also need to be consulted, as some may not want a fixed link.
"It's not an imminent need," he said. "I think there's more of a need in Orkney than Shetland in terms of the life expectancy of the ferries.
"But given the timescales of building these sorts of projects, it's good to start the discussions now and explore the different options."
He added: "It's one of these things I'm coming to with a completely open mind. I think the experience in the Faroe Islands shows it can be done."
Sigurd Lamhauge, chief executive of Landsverk, the organisation which oversees transport projects on the Faroes, said: “The system of tunnels we have built in the Faroes has led to increased incomes, improved access to healthcare and education, better commuting times and resulted in more people staying on the islands.
“They have brought positive change to the Faroes and I would think something similar would have the potential to do the same to places like Orkney, Shetland and other Scottish islands.”
The most recently completed tunnel in the Faroes, Eysturoyartunnlin, connects the islands of Streymoy and Eysturoy and has cut the travel time between Torshavn, the capital, and the village of Runavik from an hour and 14 minutes to just 16 minutes.
A spokesman for SNP transport minister Jenny Gilruth said: “If Mr Stewart is interested in sharing any of the facts he has gathered from his trip, the Scottish Government would be more than willing to hear them, recognising, as Mr Stewart will of course respect, that transport is devolved.”