The House began debating the measure Tuesday afternoon.
Republicans who control both the House and Senate have called the bill a top priority for the year. Opponents argue it will encourage racial profiling.
The bill specifies that the check must follow a stop or arrest for something else. Unlike a law passed in Arizona last year, it says people can't be held on the suspicion. Instead, they must be processed normally if authorities don't respond.
The measure would also make it a felony to make fake photo IDs for illegal immigrants.
Senators passed their own version earlier this year.