The culture secretary has said she is "not ruling out" changes to the government's Online Safety Bill in the face of a major backbench rebellion.
Forty-three Tory MPs are backing a plan to make social media bosses face prison if they fail to protect children from damaging online content.
In the bill's current form, firms would instead face higher fines.
Michelle Donelan told the BBC's Newscast she would take a "sensible approach" when considering MPs' ideas.
Under the proposals, senior managers at tech firms could face up to two years in jail if they breach new duties to keep children safe online. The provision would not apply to search engines.
Currently the bill would only make managers criminally liable for failing to give information to media regulator Ofcom, which is set to gain wide-ranging powers to police the internet under the new law.
Among the rebels backing the amendment, which is due to be voted on next week, are former Cabinet ministers Priti Patel and Dame Andrea Leadsom.
Labour has confirmed to the BBC it also supports the amendment, meaning rebel MPs could have the numbers to inflict the first defeat of Rishi Sunak's premiership if it is called to a vote.
Miriam Cates, a leading Tory rebel, told the BBC the group of backbenchers met Ms Donelan earlier this week, and ministers recognise the "strength of feeling" over the issue.
She added that they were open to government concessions, but any proposal to change the law would have to retain personal liability for managers.
"I think that is the key driver of change," she told the BBC's World Tonight programme, adding: "In the construction industry we've seen a massive drop in accidents and deaths in construction since the senior manager liability was introduced."
Mr Sunak has already faced significant backbench revolts in the first two months of his premiership – over housing targets for councils and restrictions on onshore wind farms.
On both of those issues, the prime minister backed down and offered concessions to avoid defeat in the House of Commons.
The Online Safety Bill was introduced in March under Boris Johnson, and has been repeatedly altered during its passage through Parliament.
Its progress was further delayed last month when the government decided to make more changes to the bill.
It is due to return to the Commons next Tuesday, before what is likely to be a long journey through the House of Lords.
Sunak facing Tory revolt over internet safety law
Band invited to speak to MPs about online safety
Tech firms accused of prioritising profits over children
Classified documents found at Mike Pence's home
Seven dead in another California mass shooting
'Joyous celebration turned into tragedy and fear'
Oscars 2023: The nominations in full
How to read the Doomsday Clock
Men must stand up for Afghan women – professor
Hero who disarmed gunman had never seen a real gun
Eight-year-old Indian diamond heiress who became a nun
What you need to know about Nigeria's elections
The Englishman who helped plot Morocco's World Cup run
What’s the future of India's sinking Himalayan town?
The bacteria controlling your brain
The simple error that 16% of us make
Gen Z's latest surprising obsession
A return to old-school Canadian glamour
© 2023 BBC. The BBC is not responsible for the content of external sites. Read about our approach to external linking.
