WASHINGTON — Officials in the Democratic Party are wooing Elizabeth Warren to run for the Senate against the Massachusetts Republican Scott P. Brown rather than have her continue to set up the new Consumer Financial Protection Bureau.

Ms. Warren has become a lightning rod for controversy over the new agency, which she conceived and is helping create. Consumer groups and some Democrats have demanded her appointment as its first director. A group of 44 Senate Republicans, with applause from the financial industry, has promised to block any nominee.

In seeking to enlist Ms. Warren for a different campaign, Democrats are taking aim at two birds. They can lay the groundwork for a potential compromise over a different candidate to lead the new agency and, they hope, they can increase their chances of reclaiming Mr. Brown’s seat by sending against him a woman who has won considerable acclaim and popularity among liberals for taking on the financial industry.

Ms. Warren, 61, a law professor at Harvard, has lived in Massachusetts since the early 1990s, commuting in recent years to work in Washington.