(CNN) The House Judiciary Committee is preparing to take initial steps to potentially hold former Trump campaign manager Corey Lewandowski in contempt over his refusal to answer questions at this week's hearing before the panel, multiple sources tell CNN.

But it is a process that could take weeks: First, a letter is expected to be sent to Lewandowski asking him to answer questions and warning him he can be held in contempt if he doesn't answer. Then, they may offer a contempt resolution, officially notice a committee vote and then hold a vote in committee before any floor action.

Some Democrats have been irked by the failure to hold Lewandowski in contempt immediately during the hearing, according to Democratic sources. That flies in the face of current House rules that would have made the process quite cumbersome to immediately hold him in contempt at the hearing.

At a private meeting Wednesday, House Speaker Nancy Pelosi said the committee should have immediately held Lewandowski in contempt, but doing so would have been an arduous undertaking under House rules. Her office says that Pelosi was merely expressing frustration at Lewandowski -- not the committee.

Corey Lewandowski, the former campaign manager for President Donald Trump, testifies to the House Judiciary Committee on Tuesday in Washington.

But the fight over what to do after Lewandowski's testimony showcases the tensions that have been building within the Democratic caucus over the committee's handling of impeachment , with members of the party again at odds over their messaging and strategy during a crucial time facing their majority. In the aftermath of the hearing, some Democrats are openly questioning the wisdom of bringing forward Lewandowski, whose heated exchanges with Democrats and stonewalling of their questions may have overshadowed key aspects of his testimony.

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