Tweeters have been using #NoToSocialMediaBill to campaign against the proposal.
It proposes up to a seven-year sentence or $25,000 (£16,000) fine for anyone found to be sending "abusive messages".
Human Rights Watch condemns the move as an attempt to muzzle free speech.
Millions of social media users in Nigeria, as well as those sending text messages, could be affected, it says in a statement.
The bill does not define "abusive statement or messages."
There will be a public hearing on the bill before it can be passed.
Activists see the bill as an attempt to target critics of lawmakers and politicians.
The BBC's Nasidi Adamu Yahya in the capital, Abuja, says Nigerian MPs often come under the media spotlight because of the huge money they earn.
However, Senator Bala Ibn Na'allah of the governing All Progressive Congress, who sponsored the bill, said the publication of false stories was becoming rampant in the country.
"You can't write false stories just because it is social media," he told the BBC Hausa service.
The offences the proposed bill seeks to criminalise already exist under Nigerian laws including those on treason, defamation, and libel, our reporter says.
Nigeria has a vibrant civil society, with many activists who use social media for their campaigns. It has the largest number of mobile phone users in Africa.

Roslyn Condie, of Mossdale, near Castle Douglas, admitted being the owner of a dog which was dangerously out of control last September.
A court heard how her Staffordshire Bull Terrier, Nipper, bit the postman on the leg leaving a puncture wound.
Sentence was deferred on Condie for six months.
Dumfries Sheriff Court heard how the incident happened while the postman was delivering mail in the village of Mossdale.
Local residents gave him first aid and he later had medical treatment for the bite.
Fiscal depute Jennifer McGill said there was concern in the area over the dog which had on occasion escaped from the garden of its home.
A solicitor for Condie said she had had the dog since it was a puppy and it was very affectionate.
She now kept it muzzled when taken out and the garden fences had been made escape proof.
Sheriff Brian Mohan handed out the warning that a second bite would bring destruction and deferred sentence on Condie stressing that she should keep the dog under control with appropriate measures.

According to the military, militants now control only small pockets within the southern Philippine city.
But there are still reports of fighting on the ground and thousands of civilians trapped.
Nineteen civilians are known to have died. Some were found dead in a ravine with gunshot wounds to their heads.
Militants allied to the so-called Islamic State came out onto the streets of Marawi last week after the army attempted to capture a top militant leader.
Many residents fled the city as clashes erupted, but a local official said about 2,000 people were unable to leave areas held by the militants.
Spokesman for the Philippines' Armed Forces Brig Gen Restituto Padilla said troops were in "complete control of the city except for certain areas" held by militants from the Maute group, the local militants who have declared allegiance to IS.
He said that "around 40 to 50 armed elements" were still present in the city, but this may increase given the militants' activities on the ground including the freeing of more than 100 prisoners from a local jail.
He added that 18 military and police personnel had been killed and more than 61 Maute fighters.
Reports say the remaining fighters are still believed to be holding several hostages, including a priest and a number of Christians.
The violence began when the army failed in its attempt to capture Isnilon Hapilon, believed to be the main IS leader in the Philippines and linked to the Maute.
In response the Maute swarmed the city, taking over a hospital and burning down buildings.
President Rodrigo Duterte then declared martial law on the southern island of Mindanao, where Marawi is located.
The Philippines, which is majority Catholic, has faced Muslim separatist movements for decades in Mindanao with its significant Muslim population.
Marawi is known as "Islamic City" in the Philippines for its Muslim-majority population.
Nigerians have been reacting angrily to a draft bill being discussed in the Senate which aims to punish anyone who "propagates false information" on electronic media.
[NEXT_CONCEPT]
A sheriff has warned the owner of a dog which bit a postman delivering the mail that her pet will be destroyed if it bites someone again.
[NEXT_CONCEPT]
The Philippine military says it has made gains retaking Marawi city from Islamist militants amid clashes that have left about 100 people dead.