White House aides said Monday that President Bush hopes by Christmas to find a nominee to replace Tom Ridge as Homeland Security secretary after Bernard Kerik (search) withdrew his nomination.

Kerik apologized over the weekend for an immigration problem involving a family housekeeper that forced him to withdraw. "I owe the president ... a great apology that this may have caused him and his administration a big distraction," the former New York Police commissioner said in a telephone interview with The Associated Press.

Speaking from his home in Franklin Lakes, N.J., the 49-year-old Kerik said he had discovered a few days ago that he did not pay all required taxes for a family housekeeper and that the woman may have been in the country illegally.

Former New York Mayor Rudolph Giuliani (search) offered apologies for the abortive nomination of his close associate during a preplanned dinner with Bush at the White House on Sunday, a Giuliani spokeswoman said.

Kerik was Giuliani's police commissioner and more recently a business associate.

"The president was very gracious," said Giuliani spokeswoman Sunny Mindel in New York. "They remain good friends."

White House spokesman Scott McClellan told reporters Monday that, "the president had a very good conversation with the mayor last night" and that Giuliani and his wife rode in the presidential limousine with the president and Mrs. Bush after they attended a friend's holiday dinner.

"The mayor indicated he made an apology to the president. I think the president thought none was necessary," McClellan said.

Earlier Sunday, Bush and Giuliani attended the taping of the annual "Christmas in Washington" concert, to be televised on the TNT cable network Dec. 15.

Who Will Get the Job?

The big question now is who will be nominated to oversee the sprawling agency, created after the Sept. 11, 2001 attacks.

On Saturday, Giuliani, who has been mentioned as a possible choice, expressed no interest in the job. "I am not a candidate," he told reporters in New York.

Sen. Susan Collins, the Maine Republican who heads the Senate committee that will take up the nomination, said two "terrific choices" would be Asa Hutchinson (search), the department's undersecretary for border and transportation security, and Sen. Joe Lieberman (search), D-Conn.

"Joe Lieberman is a terrific person and has the faith and confidence in everyone in the Senate and at the White House, quite frankly," said former White House political director Ron Kaufman.

But Steve Murphy, who served as the campaign manager for former presidential hopeful Dick Gephardt (search), said there could be political repercussions to tapping Lieberman. Even though the senator is a Democrat, members of his party might balk because Connecticut has a Republican governor who would be responsible for picking a replacement should Lieberman leave the Senate.

"I think what matters here is the next homeland security secretary … there shouldn't be any partisanship whatsoever in the choice," Murphy added. "Democrats and Republicans have got to join together to do the job as best we can."

Among the other names that had been circulating for the post before Kerik's selection on Dec. 3 were Joe Allbaugh (search), former director of the Federal Emergency Management Agency (search), and White House homeland security adviser Fran Townsend (search).

But it was still possible that the White House would break the search wide open again rather than return to the previous stable of top contenders, said a senior administration official.

Vetting Process Plagued?

Meanwhile, questions continue over how the White House review process could have missed the kind of "nanny problem" that scuttled high-level appointments in both the Clinton administration and the first Bush administration.

"There is a thorough vetting process involved when the president intends to nominate someone. Our vetting process looks at all the issues related to a potential nominee's personal, professional, and financial background, and we did this in this instance, as well," McClellan said Monday.

Lawyers from the White House counsel's office also spends time with that candidate, McClellan said, nothing that, "in that process, we do rely on that candidate to provide us with all the information necessary."

He added that the White House is "moving forward with finding a replacement" for Kerik.

But some lawmakers aren't satisfied with the current vetting process.

"I'm not sure how much background checks were done before the announcement," said Louisiana Sen. John Breaux, a Democrat. "Sometimes it's difficult to have those things completed before they make their announcement. But when you don't have it done before, you run into these type of situations."

Giuliani said the president's aides did all they could.

"They did everything that they could do. They asked the right question, same question I asked many, many times when I did this. They asked him, 'is there any complications with your domestic situation,' you know, domestic workers or something like that."

Giuliani said Kerik could never have survived the Senate confirmation hearings. As homeland security director, Kerik would have run the immigration department.

Giuliani said Bush administration officials asked Kerik from the beginning whether there were any issues involving domestic employees.

"He didn't think he had a problem," Giuliani. "He made a mistake."

But others said the back-end process wasn't to blame.

"This all is on Bernie Kerik, I don't think it's on the vetting process," Murphy said. "I really don't think it's fair to blame George Bush for that."

Bush, leaving a Maryland hospital after his annual physical Saturday, ignored a question about Kerik's withdrawal.

Bush advisers, now having to deal with the kind of messy situation they had so far avoided, were taking Kerik at his word that he had not intentionally misled them, the official said.

Because the issue involved immigration — and the Homeland Security Department includes the immigration agency — Kerik had no choice but to withdraw, Giuliani said. "Every time immigration issues came up this would be a problem."

"It would have been a bitter, difficult battle that probably would have ended without him getting confirmed," Giuliani said.

In an AP interview, Kerik said that on Wednesday — five days after Bush announced his selection — he discovered financial records "that led me to question" whether proper taxes had been paid for the housekeeper. By Friday, he said, "I came to realize ... there may have been a question with regard to her legal status in the country."

Some lawmakers have also questioned whether Kerik had the management experience to oversee the department, which has 180,000 employees.

Democrats also focused on potential conflict of interest issues because of Kerik's recent $6.2 million windfall from exercising stock options in a stun gun company that does business with the federal department.

According to a report in Newsweek, Kerik also was involved in a civil dispute six years ago after he failed to pay maintenance fees on a New Jersey condominium he owned. At one point an arrest warrant was issued, the magazine said.

Kerik on Saturday denied an arrest warrant was ever issued, but said he had been ordered to appear at a foreclosure hearing.

FOX News' Liza Porteus and Wendell Goler and The Associated Press contributed to this report.