The social media giant shows no sign of backing down after banning access to news and information pages across Australia in response to the code.
Mr Frydenberg said there had been no movement overnight but another meeting was scheduled on Friday.
In text message exchanges with Mr Zuckerberg, he expressed disappointment at Facebook's "unnecessary and heavy-handed" blackout of Australian news and information.
The treasurer said the purpose of the bargaining code extended far beyond paying fairly for journalism.
"There is more at stake here than just one or two commercial deals," he told ABC radio.
"This is very much about Australia's sovereignty, this is about Australia making laws for Australians, this is very much about the rules of the internet and the digital world replicating the rules of the physical world."
Mr Frydenberg is calling on other nations to back Australia in its quest to force digital giants to pay for locally produced news published on their platforms.
Opposition Leader Anthony Albanese said the decisions to block emergency service sites was shocking.
"Facebook's behaviour has been reprehensible," he told the ABC on Friday.
"They've done a great deal of damage to their reputation and businesses rely upon their reputation."
Mr Albanese said the company needed to accept media companies should be paid for content to keep journalism alive.
"It's in Mr Zuckerberg's interests, and Facebook's interests, to basically grow up, to accept that governments have a right to determine the regulatory framework in which businesses operate."
Prime Minister Scott Morrison described Facebook as arrogant for shutting down emergency services and community pages in its Australian blackout.
"Facebook's actions to unfriend Australia today, cutting off essential information services on health and emergency services, were as arrogant as they were disappointing," he said in a statement on Facebook.
He said Facebook's actions confirmed global concerns about the behaviour of big tech companies.
"They may be changing the world but that doesn't mean they run it."
Facebook claimed it had no choice but to shut health and emergency services pages down, arguing the bargaining code was poorly worded.
"However we will reverse any pages that are inadvertently impacted," the company said in a statement.
Facebook gave no notice of the news ban but it was not unexpected.
The company first threatened to ban news for Australians in August and repeated the ultimatum before a Senate inquiry in January.
The ban restricts Australian users and publishers from viewing or sharing domestic and international news.
Overseas users will be unable to access Australian news.
The media bargaining code is before the Senate after clearing the lower house.
It is likely to clear parliament with bipartisan support, despite Labor criticising the government for its handling of negotiations with digital platforms.
Comments
Australia's attitude is liking trying to get more money out of tradesperson
I really fail to see the problem here.
Somehow I have managed to live without Facebook in my life, ever. I may well use some of the apps linked to its base, like Messenger or What's App but not having Facebook itself would not impact my life.
I'd welcome a similar ban here in NZ.