WASHINGTON, D.C.: Giving himself some running room as a Senate vote on late-term abortions looms, President Clinton announced he might throw his support behind a slightly different version of the bill he vetoed last year. "Since the veto, Clinton and the Democrats who supported him have been knocked about politically,” reports TIME’s James Carney. “The procedure is something which appalls a majority of people and which they oppose, unlike abortion in general, where a majority supports abortion rights. By backing the new bill, the President and his allies in Congress get a lot of cover politically because it suggests they may be serious about ending the practice to an extent." The new measure, sponsored by Minority Leader Tom Daschle, was shaped in response to the gruesome partial birth abortions that the GOP wants to ban except when the mother's life is endangered. The Daschle version would also allow the procedure when giving birth would risk "grievous injury" to the mother’s health. Republicans say that addition leaves the bill no different than the current law, which effectively permits women broad license to abort a fetus even in the eighth month of pregnancy. As Republicans search for the six votes needed to pass their tougher measure with a veto-proof majority, many Senators remain publicly undecided. They are scheduled to vote by week's end.