Launched on 20 May, it attracted 15,000 signatures in its first four days. This was already more than half of the 25,000 signatories that must be obtained within 30 days in order to trigger an official response from the US government.
The petition says that expanding access to research would "speed the research process and increase the return on our investment".
It argues that the "highly successful" open-access mandate that has already been adopted by the US' National Institutes of Health "proves that this can be done without disrupting the research process".
The petition has been launched by a group of advocates under the banner "Access2research".
On its website, the group claims that there is a "serious debate" taking place in the White House about open access, but it believes a further push is needed.
"The goal is not just to get 25,000, but [also] to blow it out of the water...to show the White House that this issue matters to people, not just a few publishers," Access2research says.
It adds that its lobbying efforts are being opposed by "a small set of publishers [that] profit enormously from the existing system" and who "can - and do - outspend those of us who have chosen to make a huge part of our daily work the expansion of access to knowledge".
The group cites as an example of publishers' power the abandoned Research Works Act, proposed in the US Congress earlier this year, which would have outlawed governmental open-access mandates.
Support for the bill offered by some major publishers prompted thousands of academics to pledge to boycott the largest of the companies, Elsevier, which later withdrew its support for the legislation.