Hundreds of Web sites produced by people who remember when Pennsylvania Sen. Arlen Specter (search) opposed the nomination of Robert Bork (search) are part of a new Internet campaign aimed to prevent the Republican from becoming chairman of the Senate Judiciary Committee.

They argue that Specter, an abortion rights (search) proponent, might actively work to block any abortion opponents nominated by President Bush to the Supreme Court. The White House and Capitol Hill have been deluged with e-mails and phone calls opposing Specter's next-in-line promotion, scheduled to coincide with the term-limited departure of Sen. Orrin Hatch from the top committee post.

Specter, who was just elected to a fifth term in the Senate, calls the stream of communications a well-orchestrated campaign.

"There are many people who would like to see me not get the position. I had a very tough primary. The same people who are out after me now came into Pennsylvania and campaigned hard against me," Specter told FOX News.

The National Review Online — daily reading in many conservative circles — published an editorial Tuesday urging that the term of the current Judiciary Committee chairman be extended.

"Senate Republicans should back [President Bush] up with the best team they can field, and that means a Judiciary Committee chaired by Orrin Hatch for two more years," the editorial reads.

Hatch of Utah has served six years as chairman, the maximum allowed under GOP rules. Speaking from Salt Lake City on Tuesday, he told FOX News that he believes other Republican senators will be reluctant to change that rule.

"We set a rule, and however dumb I may think it may be or others may think it may be, others think it's a perfectly good rule. It's the rule of the caucus and I'm prepared to live with it," he said, adding that he is not among those concerned about a Specter tenure at the top of the panel.

"My experience with Arlen Specter has been a good experience," he said.

At the White House, spokesman Scott McClellan declined to enter the fray.

"That's a matter for the Senate to decide. And it's something that the White House does not have a role in. Senators will make that decision," he said.

Senate GOP leadership could probably quell this rebellion by openly supporting Specter, but two independent Republican sources told Fox News that will probably not happen.

Specter is on his own to sell his chairmanship to party conservatives and, more immediately, the six conservative members of the Senate Judiciary Committee who will decide by secret ballot in January whether to forward Specter's name to the Republican Caucus as their choice to take over the panel.

Click on the box at the top of this story to watch a report by FOX News' Brian Wilson.