The lord advocate said offensive behaviour would include an aggravating factor Celtic and Rangers football clubs have called on the Scottish government not to rush into legislation to tackle sectarian behaviour.
MSPs are poised to pass the emergency law to increase jail terms for sectarian-related behaviour connected to football.
The Scottish government expects the bill to be law by the end of the month.
Rangers and Celtic both issued statements warning that the laws risked being flawed if rushed.
The Offensive Behaviour at Football and Threatening Communications Bill is expected to be passed in time for the new season in July, despite only being introduced to parliament last week.
The bill – which faces its first vote in parliament later on Thursday – aims to stamp out abusive behaviour from football fans whether they are watching matches in a stadium, in the pub or commenting online.
It would raise the maximum jail term from six months to five years.
A spokesman for Celtic FC said the club had always stood against sectarian and racist behaviour.
Two new offences on football-related behaviour regarded as offensive and threatening.
One deals with disorder around football matches inside the ground, and extends to those travelling to and from stadiums – as well as fans watching games elsewhere, for example in pubs or on big screens outdoors.
The second offence deals with serious threats – including murder – made on the internet.
That would take in posts on sites like Facebook and Twitter, as well as specific websites.
Both offences would become indictable, with a maximum punishment of five years in jail.
The maximum jail term for sectarian hate crimes is currently six months.
Decisions on sentencing in each case are a matter for the courts.
Offensive Behaviour at Football and Threatening Communications Bill
However, he added that the club had only four working days to consider the legislation.
“We have not, therefore, been given anything like sufficient opportunity to scrutinise the legal or practical effects of the proposed legislation or the questions and issues arising,” he said.
“Rushed legislation is rarely competent legislation. We believe it is better to have it right rather than rushed, and it is clear that this view is shared widely by many others.”
A Rangers spokesman said: “We are concerned that in the desire to see new laws on offensive behaviour in place in time for the new football season, there is a real risk being created that the legislation could be flawed and ultimately not achieve its objectives.”
He said it was important for fans to have their say on the bill.
The spokesman said the timescales were “extreme and may be counter-productive”.
“We hope as a club these concerns are taken on board and everyone can continue to work together to ensure legislation can be effective and durable,” he said.
On Wednesday, two Christian organisations decided to take legal action to try to force the Scottish government to take more time over the legislation.
The Christian Institute and Care for Scotland expressed concerned that the bill would curtail ordinary religious debate, and are hoping the courts will rule the legislation is unlawful.
The Scottish government said it did not regard the legal challenge from the Christian groups as being valid.
The new legislation comes in the wake of several high-profile football-related incidents.
These include trouble at Rangers/Celtic games and the sending of suspected bombs to Celtic manager Neil Lennon and two other high-profile supporters of the club.
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