As Republicans across the country mount an aggressive effort to tighten voting laws, a group of former aides to President Obama and President Bill Clinton is pledging to counter by spending up to $10 million on a push to make voter registration automatic whenever someone gets a driver’s license.

The change would supercharge the 1993 National Voter Registration Act, known as the “motor voter” law, which requires states to offer people the option of registering to vote when they apply for driver’s licenses or other identification cards. The new laws would make registration automatic during those transactions unless a driver objected.

The group, called iVote — which is led by Jeremy Bird, who ran Mr. Obama’s voter turnout effort in 2012 — is betting that such laws could bring out millions of new voters who have, for whatever reason, failed to register even when they had the opportunity at motor vehicle departments.

Many of those new voters would be young, poor or minorities — groups that tend to support Democratic candidates, Mr. Bird said.