"I think anyone with common sense would be reluctant to go talk to this committee," the GOP congressman said.
The cartoon pictures the GOP congressman as a dog, with a red tie functioning as a leash being held by Trump's hand.
The committee has stated that Jordan "[failed] to address the principal bases for the Select Committee's request for a meeting."
The Ohio representative and staunch Trump ally confirmed he will not voluntarily speak to the House Select Committee investigating the riot.
The top Republican congressman on the House Judiciary Committee retweeted the since-deleted post.
Swalwell blamed "Republican liars" for the consequences of the prolonged COVID-19 pandemic.
The Ohio representative made the comments during an appearance on Fox News' "Sunday Morning Futures."
Host Jonathan Capehart warned that "all hell's going to break loose" if the Ohio Republican becomes House Speaker.
Representative Madeleine Dean called Jordan not a "serious American" and someone who doesn't "care about democracy."
The Republican accused the House Select Committee of misleading the American people with a truncated text message.
The Ohio representative has repeatedly recounted disproven claims of voter fraud in the 2020 presidential election, which helped cause the Jan. 6 riots.
The Illinois Republican said the House select committee would "absolutely" be willing to subpoena members of Congress and former President Donald Trump.
Attorney Glenn Kirschner said Saturday "there's probable cause that Jordan committed a crime by forwarding that text to Mark Meadows."
The January 6 committee chair has been accused of "doctoring" the content of a text message sent to Mark Meadows by the Ohio congressman.
Jordan's office said he had forwarded Meadows a message about rejecting Electoral College votes on January 6.
"All electoral votes that [Mike Pence] believes are unconstitutional" should've been discarded, according to the text.
The Ohio Republican's message on Twitter was met with a large response, especially from critics.
Once the House's No.3 Republican, Cheney's opposition to former president Donald Trump has made her the focus of GOP ire.
"I hope that none of the Republicans abuse executive privilege. This is about having a private citizen answer for their role on January 6," said Rep. Ro Khanna.
Gaetz cautioned against congressional Republicans developing "impeachment fever" if they regain control of Congress in 2022.
The Ohio Republican and the infectious diseases expert have clashed several times over COVID-19 prevention measures.
Social media users and fellow members of Congress mocked the GOP congressman's suggestion.
If they were to go head-to-head in 2024, the former president is slightly favored with 47 percent against Biden's 46.
"How can it be an extraordinary success and at the same time be President Trump's fault?" Jordan asked Sunday.
"Everything this guy touches, it goes bad. That's why I said he should resign," said the Ohio Republican.
Jordan said "President Trump, he's gonna run again" despite a spokesperson denying he said something similar.
Representative Jim Jordan (R-Ohio) reportedly said that Trump will announce his intention to run in 2024 "any day now."
The Republican congressman admitted last weekend that he spoke with Trump "more than once" during the Capitol attack.
"I know it was more than once," the GOP congressman said of his calls with Trump on January 6.
Republican House leaders are arguing that the bipartisan committee should focus more on whether Speaker Nancy Pelosi should have done more to secure the Capitol.