The Rules Committee approved the measure in a 9-4 vote along party lines.
The resolution was drafted and first approved on Tuesday by the US House Select Committee investigating the events at the Capitol on January 6.
Contempt of Congress charges can only be pursued by the Justice Department and can result in a penalty of up to one year in prison and a fine of up to $100,000.
US media reported the full vote could take place on Thursday.
Last week, Bannon refused to comply with a subpoena that required him to testify before the Select Committee and referenced the executive privilege rule in his defense. However, the committe sent a private letter to Bannon’s lawyer in which it rejected his arguments not to cooperate in the ongoing probe and threatened to hold him accountable for contempt of Congress.
On Monday, former US President Donald Trump sued the Select Committee for what he says are illegal requests to obtain White House records, saying the records have no connection to the events. Trump’s lawsuit points out that the FBI has found no evidence the event was part of an organized plot to overturn the 2020 election results nor that Trump and his associates were involved in any such exercise.
On 6 January 2021, thousands of Trump supporters stormed the Capitol building in a bid to stop Congress from certifying Joe Biden's win in the November election. The storming was preceded by a Trump rally, which saw a massive turnout. Five people, including a Capitol Police officer, died as a result of the riot.
The president, who has repeatedly claimed the 2020 presidential election was “stolen” from him, was accused of “inciting an insurrection". Trump vehemently rejected the accusations, spearheaded by the Democrats, that he had instigated the riots. He was later impeached by the US House of Representatives on charges of incitement to insurrection, but was subsequently acquitted by the US Senate.