Massachusetts drivers would have to keep their hands off their cellphones — for talking and texting — under a bill approved Thursday by the state Senate.

The proposed legislation would amend the oft-violated 2010 ban on texting behind the wheel by barring drivers from using their hands to make calls. Drivers could still talk on the phone, but only if they used Bluetooth or other technology that allows for hands-free calls.

The Senate’s proposal would also explicitly ban the use of social media and video calling systems while driving. It would prohibit drivers from using their fingers to type addresses into GPS systems and mapping apps, though they could still display such apps to help with directions.

Drivers could use their hands for “a single tap or swipe’’ to start or end hands-free calls, and could do the same to activate GPS programs. But drivers would be subject to a citation for even holding their phones near their heads or on their laps.

The distracted-driving proposal now moves to the House, which has ordered a similar bill to a third reading, one of the final steps before a bill comes to a vote. Seth Gitell, a spokesman for House Speaker Robert DeLeo, declined to say whether the House would bring the bill to a vote this session.

“The bill will be reviewed as it goes through the process,’’ he said.

Billy Pitman, a spokesman for Governor Charlie Baker, said the “administration shares the concerns many have about distracted driving and public safety’’ and “will carefully review any final legislation reaching the governor’s desk.’’

So far, 14 states require drivers to use hands-free technology to make calls, according to the Insurance Institute for Highway Safety, which researches roadway safety. The states include Connecticut, Vermont, New Hampshire, and New York.

The Massachusetts bill calls for fines of $100 for a first offense, $250 for a second offense, and $500 for a third or subsequent offense within six years — similar to the penalties imposed under the existing texting law.

A third or subsequent violation could also result in an insurance surcharge.

On Thursday, some state senators criticized the fines as too burdensome on low-income individuals. But a push to lower them fell short.

Senators also described the hand-held ban as a way to strengthen the state’s 2010 texting law, often criticized as difficult to enforce.

Under the current law, drivers are allowed to dial phone numbers, which opponents say forces them to take their eyes off the road and serves as a ready-made excuse for texting.

“Anybody who drives on a regular basis in Massachusetts can see, on a daily basis, people texting,’’ state Senator Eileen Donoghue of Lowell said.

Such drawbacks led to one of the most heated points of debate Thursday: Whether drivers should be cited for keeping phones on their laps, where they might be suspected of trying to dash off a text or e-mail.

In the end, lawmakers approved a provision that drivers who rest a phone in their hand and on their lap could be fined.

Versions of hands-free legislation have come up in the Senate for more than a decade, since well before the texting ban, state Senator Mark Montigny of New Bedford said.

“Texting wasn’t even an issue,’’ Montigny said. “We were trying to take the physical product out of the hands of people.’’

Adam Vaccaro can be reached at adam.vaccaro@globe.com. Follow him on Twitter at @adamtvaccaro.