Around the clock and round the clock are both common forms of the phrase meaning all day or at all hours. Round is a variant of around—though this use of round is more common in British English than in American English—and the words are interchangeable in this expression.
Round the clock is more often used as a phrasal adjective preceding a noun. In these cases, it should be hyphenated—e.g., round-the-clock monitoring, round-the-clock activities. Around the clock appears more often as an adverbial phrase, usually following the verb it modifies. In these cases, around the clock should not be hyphenated—e.g., activities taking place around the clock.
These writers use round-the-clock as a phrasal adjective:
But after three years of round-the-clock work by 3,000 Irishmen, the liner was going into the water all the same. [Washington Post]
In January, Winfrey launched her own TV network, which will offer round-the-clock lifestyle programmes. [BBC News]
And these writers use around the clock adverbially:
So, it’s something you have to do around the clock, and that doesn’t compute with our existing educational system. [transcribed on NPR]
AI applications can monitor communications around the clock to hunt suspects or detect terrorist plots. [Globe and Mail]
In British English, where round is often used in place of around, round the clock is often used adverbially—for example:
Chinese and German scientists have been working round the clock to identify the strain. [The Sun]
As it happens, Mr Murakami employed more than 200 craftsmen working round the clock. [Economist]
And around the clock is occasionally used as the phrasal adjective—for example:
After emerging from a coma he was discharged from hospital but would require around-the-clock care. [The Age]
The task force also recommends maintaining funding for around-the-clock staffing at domestic violence shelter. [Wall Street Journal]