The bill would allow sales to people over the age of 21 and establish a licensing process for deliveries to customers.
New Yorkers could grow up to three mature plants and three immature plants for personal use.
The penalties for having less than three ounces of the drug outside the home would be removed and the records of those previously convicted of a low level would be erased.
Democrats have a veto-proof majority New York State and have made legalizing cannabis a priority this year.
If the bill passes as planned – perhaps as early as next week – it could come into force in 2022.
Fourteen other states already allow the sale of cannabis for recreational purposes, rather than for medical purposes, but New York has so far failed to pass similar legislation.
It is estimated that an additional annual tax of $ 350 million (£ 254 million) would be generated.
Supporters say tens of thousands of jobs would also be created and that would settle a war on drugs that disproportionately targets minority and poor communities.
The bill’s sponsor in the Senate, Liz Krueger, said: “My goal in passing this legislation has always been to end the disparate racial application of the marijuana ban that has taken such a toll on communities. of color throughout our state, and to use the economic system as a bonanza of legalization to help heal and repair these same communities. ”
Grants and loans would be given to encourage people from minority communities to get involved in the newly legalized industry.
Previous efforts to legalize cannabis in New York City have failed due to factors such as how sales tax revenue is distributed and concerns about people driving at high speeds.
Others have expressed concerns that this will send mixed messages to young people.
An education and prevention program targeting school-aged children is planned in an attempt to allay some of these concerns.