Vallas told the Board of Education Tuesday night he and school officials must discuss when his 60-day notice should begin, The Connecticut Post reported.
"The clock will start ticking when the board decides it should tick," he said. "Do not underestimate my time or ability to focus my attention on Bridgeport."
Board member Maria Pereira was dissatisfied with Vallas' lack of specificity.
"To me it was a letter of resignation without an ending on it," she said. "His focus needs to be on this district."
Sauda Baraka, a school board member who won re-election on Nov. 5, asked Vallas if he would give the board a letter indicating the 60-day start date.
Vallas said that would be up to the new board, which will comprise four new members elected last week and five returning board members.
"I am trying to give them flexibility," he said.
The school board elections gave control to critics who want Vallas out. He said the election showed a desire to eliminate vestiges of state control of Bridgeport's schools and change superintendents.
Vallas was hired as superintendent after the elected school board was replaced by another appointed by the state. The Connecticut Supreme Court ruled the move illegal.
Vallas, a former Chicago schools chief, was selected last week by Illinois Gov. Pat Quinn to be his 2014 running mate.
How does adding the name Paul Vallas to the Democratic ticket help Governor Pat Quinn win re-election next year? It's a question Vallas himself could not answer.
"I was always concerned about whether or not I could help him politically," Vallas said. "I know what I can do to help the governor govern."
Quinn, who's known Vallas more than 30 years, said the choice is about familiarity.
"I decided to pick Paul as my running mate because I've known him a long time. He knows Illinois like the back of his hand," Quinn said.
"The governor made the decision to pick Paul Vallas based on his record of being a strong advocate for education reform," Chicago Mayor Rahm Emanuel said.
"Pat Quinn has had some challenges in the management department. He's had some challenges in terms of competence, at least some people believe. This helps add to his aura of knowing how to run good government," said Laura Washington, ABC7 political analyst.
But Republicans - with three women and one Asian American running for lieutenant governor in their primary - note the Democratic ticket's lack of diversity and geographic balance.
"Paul and Pat Quinn are both from Cook County. And the collar counties and downstate deserve representation, too," said State Sen. Kirk Dillard, (R), candidate for governor.
"We think that our candidates are not only good at diversity but also good at government operations but understand how to govern," said Jack Dorgan, Illinois Republican Party.
The governor said his administration includes people from around the state of all backgrounds.
"Everybody's in, nobody's left out when it comes to our administration," Quinn said.
Quinn and Vallas won't have much to do until after the March 2014 primary. The incumbent's only challenger is longshot Tio Hardiman, who still has not qualified for the primary ballot.
Last week, many politicians were stunned by the news of Quinn's choice. Vallas was the Chicago Public Schools CEO from 1995 to 2001. Since then, he has been on the road for the past decade fixing public school systems and has never held elective office. In 2010, he considered running as a Republican for Cook County Board president. He never gave up his residence in south suburban Palos Heights.
Some political consultants say Vallas would help Quinn and have an appeal in the Chicago suburbs where he ran well in the 2002 Democratic primary for governor, although his bid was unsuccessful.
The Associated Press contributed to this report.
