WASHINGTON – Three more Democrats in the U.S. Senate said yesterday that they will support the nomination of Judge Roberts as the next chief justice, increasing to 17 the number of Democratic votes the nominee will get and giving rise to speculation that the minority party has agreed on a strategy whereby it hopes to gain credibility in voting against President Bush’s next nominee by voting in favor of this one.

With nearly three-fourths of the Senate voting for Judge Roberts, attention has turned to the president’s nominee to replace Justice O’Connor. Mr. Bush’s press secretary, Scott McClellan, told reporters yesterday that the White House was close to wrapping up its consultation process with senators.

The comment suggested that a nominee could be announced as soon as tomorrow, the day Judge Roberts is expected to be confirmed.

With liberals and conservatives eagerly awaiting the name of the next nominee, the White House has kept a tight lid on its selection process.

Though conservative advocacy groups said the White House has kept the lines of communication open, Bush administration allies said that even many members of the president’s top-level staff have not been told of his choice.

Judge Roberts was assured 72 votes as of late yesterday, including those of Senators Lieberman and Dodd, both Democrats of Connecticut; and Senator Levin, a Democrat of Michigan.

Mr. Lieberman is a member of the so-called “Gang of 14” in the Senate that hashed out a plan earlier this year to end a Democratic filibuster of the president’s judicial nominees.

Mr. Lieberman urged the president to pick a replacement for Justice O’Connor who does not have “a disruptive, divisive, predetermined ideological agenda.”