(CNN) -- Democratic sources tell CNN that President-elect Barack Obama and Sen. Hillary Clinton had a "serious discussion" Thursday to gauge Clinton 's interest in the post of secretary of state, if she were offered it.
Clinton's response is unknown, although multiple sources agree that she was left with the impression that if she were interested in the post, it would be hers.
"There was a serious discussion to determine whether, if offered secretary of state, she would accept it," said a source close to the Obama transition team.
In these circumstances, that source added, a president-elect does not meet with potential Cabinet secretaries unless he is serious about making an offer.
Another source close to the Obama transition team said that asking Clinton to be secretary of state "has been of great interest to Obama for a while. You've got to assume that Hillary Clinton did not come to visit the city of Chicago ."
Over the course of the past 24 hours, sources close to Clinton have softened their one-time solid public position that she would not be interested in a Cabinet post. Those sources now say Clinton is clearly contemplating various ways in which she can serve the Obama administration.
Clinton, however, remained mum Friday about the speculation over the meeting.
Clinton went to the meeting Thursday with Obama because "she knew Obama wanted to talk about whether she would have a role in the administration," according to two sources.
Obama and Clinton met in Chicago , Illinois , at the request of the president-elect, the sources said.
On Friday, Clinton responded to the rumors at a transit policy speech in Albany , New York .
"I'm very happy there is so much press attention and interest in transit," Clinton said to laughs.
"In the off chance that you're not here for this important issue and are here for some other reason, let me just say that I'm not going to speculate or address anything about the president-elect's incoming administration."
She added, "I'm going to respect his process, and any inquiries should be directed to his transition team."
This is not the first time Clinton has been rumored to be under consideration for a position on Obama's team.
The New York senator was said to be on Obama's short list of vice presidential picks this summer, but Obama chose Sen. Joe Biden of Delaware .
One source said Clinton thought the speculation that she would be picked to be Obama's secretary of state was "silly" because of the buzz about her becoming Obama's running mate.
A spokesman for the former presidential candidate and first lady, Philippe Reines, said that "any speculation about Cabinet or other administration appointments is really for President-elect Obama's transition team to address."
On Monday night, while walking into an awards ceremony in New York , Clinton was asked whether she would consider taking a post in the Obama administration.
"I am happy being a senator from New York . I love this state and this city," she said. "I am looking at the long list of things I have to catch up on and do.
"But I want to be a good partner, and I want to do everything I can to make sure his agenda is going to be successful." Other names that have been mentioned for the top diplomatic role include Sen. John Kerry of Massachusetts , the Democrats' presidential nominee in 2004, and New Mexico Gov. Bill Richardson, a former U.S. ambassador to the U.N. and energy secretary in the Clinton administration.
Latino groups are championing Richardson for the top State Department slot. If picked, Richardson will be the nation's first Latino secretary of state.
A source close to transition team said Obama is also trying to build a diverse Cabinet that includes women and minorities and that works as a team.
Obama's transition team is expected to announce some high-level staff positions Friday…