The modified proposal would include insurance reforms, such as preventing insurance companies from denying coverage to people with pre-existing conditions, according to the source.
The potential deal would give insurance companies a defined period to make such changes in order to help cover more people and drive down long-term costs. But if those changes failed to occur within the defined period, a so-called "trigger" would provide for creating a public option to force change on the insurance companies, the source said.
Snowe is pivotal to the debate because she may be Obama's last possibility for getting a Republican senator to support his push for a health care overhaul.
She is one of the so-called "Gang of Six" members of the Senate Finance Committee, three Democrats and three Republicans, involved in separate negotiations on the only bipartisan health care proposal in Congress so far.
However, the slow pace of those talks and recent partisan attacks by the other two Republicans in the negotiations have dimmed hopes for a breakthrough, leaving Snowe as the only Republican senator that White House aides believe they can work with on the issue.
Snowe first proposed the so-called "trigger" idea for a public option months ago, and has talked to Obama about it on several occasions, according to a source familiar with the discussions.
The source told CNN that the White House staffers increased their phone calls to Snowe aides and their interest in her trigger idea this week.
Obama and Democratic leaders seek a solution that could win support from a Republican or two, and more importantly, help bridge a divide among Democrats on the public option issue.
Allies of the president hope that if Snowe accepts a health care agreement, she might also bring along her Republican colleague from Maine, moderate Sen. Susan Collins.
