House aides said Democrats’ strategy in that chamber would be affected by the margin of the Senate vote.

“If you have 70 people in a bipartisan vote, I don’t see a situation where we really play that game of chicken and change [the bill],” said a senior aide to a House Democrat supporting the legislation…

Even if the House Democratic leaders decide to push for a higher estate tax, there’s no guarantee they would be able to overcome the combined efforts of the White House and congressional Republicans. Almost all the 179 House Republicans are likely to support the current deal’s terms. That means the White House needs only about 40 Democrats to assemble a majority of 218.

The White House was working to build support among 52 House Democrats who were defeated in November, many of whom are considered centrists open to compromise on taxes.