He said it did not need to cost a lot and the recordings could be time-stamped so people knew where to go to watch items that interested them.
Cr Morris also wants an online record of how the council votes on issues - those who voted for and against.
Apart from the big issues, he did not expect many people to watch the live streaming. Neither did he expect it would lead to big increases in voting, but the council would be more user-friendly.
Mayor Stuart Crosby reacted favourably because a substantial upgrade of the council's computer system was already under way. The council was also improving the sound system in the debating chamber.
It meant that people with an interest in a certain topic could view it rather than go to the meeting: "Sometimes it is not always convenient to attend."
Ultra-fast broadband would also make it easier for users, and phone technologies meant people could watch the meetings remotely.
Inquiries by Councillor Rick Curach into Hamilton's system, which began a year ago, showed 100 to 200 people were logging on to view meetings but most soon logged off, presumably because it was boring, he said.
Cr Curach, who previously mooted the idea, said: "We would have to look at the costs ... because if only a handful of people were viewing, then it could be better to stick with the audio recording of meetings."