The National Association for the Advancement of Colored People has said any redistricting plan should at least retain the current number of 14 majority black districts.
Black lawmakers met behind closed doors with members of the NAACP to discuss the redistricting plan.
Under the current Democratic plan being considered in the House, blacks will make up the more than 50 percent of the population in 12 districts. A decade ago, blacks constituted a majority in 16 districts.
House Democrats are hesitant to draw up plans that could erode their current 62-58 majority in the House.
The issue has been debated for about a month, with each side presenting several different versions of the redrawn districts.
The first reading of the bill passed by a 62-57 vote a week ago, but two black Democrats announced on the House floor that they were not satisfied with the current plan, and several others have not said whether or not they will vote for the plan a second time.
Several black Democrats broke with their party and accused the House leaders of diluting black districts in the attempt to secure a Democratic majority. The NAACP statement has the potential to encourage further dissent.
House Speaker Pro Tem Joe Hackney, D-Orange, who serves as vice chairman of the Legislative Redistricting Committee, said it was "entirely appropriate for the plan to be re-examined" in light of the complaints.
But Hackney added that aside from the 12 majority black districts, two others are more than 49 percent black.