RICHMOND — The House again delayed action on the surprise Senate redistricting map, voting Thursday to push it off until next Wednesday.

Delegates took the issue off the front burner as the General Assembly tries to complete work on Gov. Robert F. McDonnell’s transportation overhaul and other legislation before crossover Feb. 5. Bills must pass one house or the other by that date or they die.

Because the proposed new Senate map was tacked onto a bill held over from last year, that deadline does not apply.

The move came a day after two black House Democrats signaled that they might vote for the GOP’s Senate redistricting plan. Their support could provide a measure of bipartisan backing and could help blunt criticism that Republicans muscled the map through the Senate in a way that was insensitive to blacks.

Republicans pushed it through the evenly divided chamber on Martin Luther King Day, when a Democrat who is considered a civil rights icon was away in Washington to attend President Obama’s inauguration. Taking up legislation calling for minor “technical adjustments” to House district boundaries, they amended it without public notice to revamp all 40 Senate districts.

An analysis of the Republican proposal to redraw the Virginia state Senate districts.

Dels. Onzlee Ware (Roanoke) and Rosalyn R. Dance (Petersburg) told The Washington Post they were they considering supporting the plan, which would create a new majority-black district in Southside but also disperse black voting power in at least eight other districts.

The likely outcome would be to elect a sixth black senator but also to diminish the overall power of Democrats in the now evenly split Senate.

“Here’s an opportunity where I can actually have a great impact on African Americans, yet at the same time, it doesn’t help the entire Democratic Party,” Ware said. “Should I always have to forgo the interests of black people for the good of the party? . . . I have a real dilemma on my hands right now.”

Some Democrats questioned whether the black delegates open to the map were angling for better committee assignments from Howell or eyeing the Senate district.

“Maybe you need to inquire if those persons are running for the district,” said Del. Delores L. McQuinn (D-Richmond).

Support from some black delegates — Ware said there were more but declined to give names — could make it easier for McDonnell and House Speaker William J. Howell (R-Stafford) as they decide how to respond to the map. Their votes would represent a measure of bipartisan backing and perhaps defuse some of the racial overtones surrounding how the plan was unfurled.