The proposal by the minority right-wing government, which is expected to pass and become law on Friday, would allow local authorities to ban begging from the start of summer.
An outright national ban would take effect at the start of 2015, as agreed by the two-party governing coalition and a centrist opposition group, which together form a majority in parliament.
Those found guilty of begging in municipalities where the ban is enforced would face fines and up to three months in prison.
Norway's Justice Minister Anders Anundsen, of the ruling anti-immigration Progress Party, said there was "a link" between begging and crime, notably pickpocketing.
Norwegian authorities say the capital Oslo records as many pickpocket thefts as Berlin, whose population is seven times larger.
Roma immigrants from Romania are expected to be the most affected by the new law.
Out of 194 beggars that the justice ministry identified in Oslo in 2012, just seven were Norwegian nationals, while the others were mainly Romanian citizens. The vast majority of identified Romanian beggars had a criminal record in Norway, the justice ministry said.
Opponents of the law have likened it to anti-Jewish legislation that has since been repealed.
"How can you believe that in 200 years' time this will not be seen as an attempt to stop Roma entering the kingdom?" asked Bård Vegar Solhjell of the Socialist Left Party, referring to an article of the Norwegian constitution that barred Jews from entering the country 200 years ago. Other opponents said the law would not succeed in lowering crime rates.
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"If they can no longer (beg), they will be obliged to turn even more to crime," Arild Knutsen, head of a national drug addicts association, told Norwegian television.
The ban would mark a return to anti-begging laws that were in place in Norway until 2005 and bring the country in line with legislation in other European countries such as Britain and Denmark.
Following a first vote late on Monday, the legislation will undergo a second reading on Thursday and the Norwegian king is expected to formally approve it on Friday.