Religious freedom charity Christian Solidarity Worldwide (CSW) said the new law means evangelism will become illegal and religious minorities could end up falling foul of the law.
There are concerns it could be used to target religious minorities, as occurred in the Charikot case in June 2016, when eight Nepali Christians were charged with attempting to convert children after sharing a comic book on the story of Jesus.
Human rights defenders in Nepal are calling for the bill to be amended.
According to CSW, the wording of Clause 158 of section 9 of the bill, which criminalises the 'hurting of religious sentiment', is similar to the blasphemy laws in Pakistan, which make it a criminal offence to insult another's religion.
The charity stated: "These laws are poorly defined and widely misused to settle personal scores, to target religious minorities or to further extremist agendas.
"Decades of misuse of the blasphemy laws have resulted in a situation where even voicing disagreement with these laws can lead to violence."
CSW's South Asia Advocacy, who can't be named for security reasons, told Premier's News Hour churches in Nepal have come up with creative strategies for sharing the gospel.
She said: "The Christian leadership in Nepal are planning to analyse how this law will impact them, and are coming up with a document to educate other leaders in the country on how they can still continue to share their faith in a way that respects the government's laws."
She also urged Christians to pray for an MP in Nepal who is challenging the bill before it gets the president's approval.
According to CSW, Lokmani Dhakal MP of the Janjagaran party of Nepal requested the removal of the sections criminalising religious conversion and told Parliament: "It seems very clear to me that this country when preparing the civil code has forgotten it is a signatory to international treaties that protect the freedom of religion and human rights... please don't let it be possible for the world to say of Nepal that we are the kind of nation that on the one hand signs international treaties but when making internal laws and in implementing them, does something else."
Listen to CSW's South East Asia Advocacy officer speaking with Premier's Tola Mbakwe:
