The Democratic Governors Association is creating a fund dedicated to winning races in states where governors have some control over congressional redistricting, the party’s first step in a long-range campaign to make control of the House more competitive.

Billed as “Unrig the Map,” the effort will target 18 of the 35 states in which governors play a role in redistricting, and where new congressional maps could allow Democrats to win House seats that are now drawn in a way to favor Republicans. The fund will be used for governors’ races over the next five years, leading up to the 2020 census.

Democratic officials said that they hoped to raise “tens of millions” for the effort and that they believed they could gain as many as 44 House seats if lines were more favorably redrawn in the 18 battleground states. Many of those states still have Republican-controlled legislatures, but with Democratic governors in place they could at least veto the next round of congressional maps and send the disputes to the courts.

The fund-raising effort is to be overseen by Gov. Terry McAuliffe of Virginia and Stephanie Schriock, the president of Emily’s List, a group that works to elect Democratic women who support abortion rights.