Google researchers said one of the “serious security flaws”, dubbed “Spectre”, was found in chips made by Intel, AMD and ARM.
The other, known as “Meltdown” affects Intel-made chips alone.
The industry has been aware of the problem for months and hoped to solve it before details were made public.
The UK’s National Cyber Security Centre (NCSC) said there was no evidence that the vulnerability had been exploited.
According to the researchers who found the bugs, chips dating as far back as 1995 have been affected.
Some fixes, in the form of software updates, have been introduced or will be available in the next few days, said Intel, which provides chips to about 80% of desktop computers and 90% of laptops worldwide.
Analysis by Chris Foxx, BBC technology reporter
Often when researchers discover a security problem, they share the information with the affected company so the issue can be fixed.
Typically, both parties agree not to publicise the problem until a fix has been implemented, so that criminals cannot take advantage of the issue.
This time it looks like somebody jumped the gun and information was leaked before a software fix was ready for distribution.
Intel said it had planned to share information next week, and several security researchers have tweeted that they have made a secrecy pact with the chip-maker.
That leaves the company in an uncomfortable situation, with a widely-publicised problem made public before the fix is ready to go.
Microchips are the basic electronic systems behind many devices such as computers and mobile phones.
In order to do their work, they must move data around, using different types of memory to temporarily store it.
In many cases, that information is supposed to be secure from attempts to snoop on it, but these two bugs mean that it could in fact be accessed by a third party.
The first reports suggested that a bug affected solely chips made by Intel, but it has since emerged that a separate flaw, Spectre, has been found in Intel, ARM and AMD chips.
“Many types of computing devices – with many different vendors’ processors and operating systems – are susceptible to these exploits,” said Intel.
ARM said patches had already been shared with its customers, which include many smartphone manufacturers.
AMD said it believed there was “near zero risk to AMD products at this time”.
On a conference call for investors, Intel said researchers had shown that hackers could exploit the Meltdown vulnerability, gaining the ability to read memory and potentially access information such as passwords or encryption keys on devices.
Microsoft, which uses Intel chips said it would roll out security updates on Thursday, adding it had no information suggesting any compromised data.
Apple is working on updates for its laptops and desktops.
As for Spectre, however, experts expect that it will be much harder to patch and none has yet been made widely available.
Google published a blog detailing what some customers may need to do. It said Android phones with the latest security updates were protected, and that Gmail was safe. It will be releasing security patches for users of older Chromebooks, while there will also be a fix for users of the Chrome web browser.
The NCSC said it was aware of the reports of the potential flaw and advised that all organisations and home users “continue to protect their systems from threats by installing patches as soon as they become available.”
Experts advised caution on the issue.
“It is significant but whether it will be exploited widely is another matter,” said Prof Alan Woodward, from the University of Surrey.