But critics say the maps violate two new state constitutional amendments.
The House is beginning floor debate on Thursday with a final vote expected Friday. The plans then return to the Senate, which already has passed its own set of congressional and legislative maps.
Three groups in the Fair Districts coalition, which sponsored the anti-gerrymandering amendments, contend the maps drafted by both chambers are defective.
The maps’ Republican sponsors deny those allegations, so the issue is likely to be decided in court.
A final vote in the Senate could come quickly. The House plans include a proposed compromise congressional map. House and Senate leaders also have agreed to accept the map the other chamber has drawn for itself.