The review was set up to look at devolution 10 years on.
Powers on speed and drink-drive limits as well as air weapons could be handed to Scotland, the Calman Commission devolution review is expected to say.
The BBC has learned the body's final report will also recommend MSPs take charge of setting income tax, stamp duty and air passenger duty.
The commission, voted in at Holyrood by Scotland's main opposition parties, has also been supported by UK ministers.
The review is to deliver its final report on Monday.
It is also expected to back borrowing powers for the Scottish Government - and will also suggest some devolved responsibilities, such as food standards labelling, are handed back to Westminster.
They are significant tax powers for the Scottish Parliament - they are not the tax powers that those who support independence would favour.
But it is a recognition, basically, that the current power to vary taxation in the Scottish Parliament hasn't worked.
Down the road a bit, the Treasury, I think, will expect a review of need across the UK - they will then argue Scotland's spending levels should be reduced.
The commission, chaired by Sir Kenneth Calman, was established to look again at Scotland's parliament, 10 years on.
It was backed by Labour, the Conservatives and the Liberal Democrats at Holyrood, although the Scottish Government has criticised the body, because it is not looking at the issue of independence - and has ruled out Westminster handing full financial powers to Scotland.
Former Scottish Labour leader Wendy Alexander, who was key to getting the Calman Commission set up, described Holyrood currently as "something of a pocket money parliament".
"The grant comes from London - the only thing that we really decide at the moment is some influence over council tax and some influence over non-domestic rates, but we don't have to make any decisions about our spending and that is simply wrong," she told BBC Scotland's Politics Show.
"If you move to a position where the Scottish Parliament has control over - as mooted in the press - half the income tax revenues in Scotland, it means that, in future, there would be a Budget in Scotland - a Scottish finance minister would be standing up and making a decision on whether to raise or not to raise up to five or six billion pounds in revenue."
The SNP has its own constitutional review - the National Conversation - currently under way.
Responding for the Scottish Government, Justice Secretary Kenny MacAskill said he would welcome new powers over air weapons and drink drive limits - issues on which he has long campaigned.
But he insisted the people had to be given their say on Scotland's future in an independence referendum - something the minority SNP administration does not currently have enough parliamentary support to hold.
Mr MacAskill told the programme: "I think it's been made quite clear what we're looking to do is for Scotland to take responsibility and to have control of our own financial matters - to have full fiscal autonomy, not just increased pocket money.
"Oil's worth £30bn per annum to this country at the present moment - we're facing £500m of cuts from the Westminster government - give us control of our oil and gas revenues and we'll take responsibility for making this a better place."