Rep. Jim Jordan (R-Ohio), the head of the renegade conservative Freedom Caucus, predicted Sunday that his group would "look favorably" on Rep. Paul Ryan (R-Wis.) if Ryan decides to run for speaker.

Speaking on "Fox News Sunday," Jordan pointed out that his group — which has a few dozen members — has endorsed Rep. Daniel Webster (R-Fla.) for speaker, but he called Ryan a "good man" and a "great communicator — the kind of messenger our party needs."

If Ryan jumps in, Jordan said, "I think our group would look favorably on him."

Jordan's comments came amid an active push by top Republicans to get a reluctant Ryan to run for speaker after Majority Leader Kevin McCarthy (R-Calif.) surprised his colleagues last week by announcing that he will not run.

Rep. Raúl Labrador (R-Idaho), also a Freedom Caucus member, called Ryan "a friend" and said that if new candidates enter the race, the caucus will evaluate them. Both Labrador and Jordan said the next speaker should embrace a new process that takes the input of rank-and-file members into greater account.

"It's not about the who, it's about the what," Labrador said on CNN's "State of the Union."

Speaking on the same program, Rep. Tom Cole (R-Okla.), an ally of top House GOP leaders, expressed hope that Ryan would come around to running for speaker. But Cole added: "He's got a lot of things to consider."

Former speaker Newt Gingrich told Fox News that Ryan should carefully consider the pros and cons of pursuing the speakership amid hot-button policy debates in the GOP conference, citing the conservative resistance that hamstrung outgoing Speaker John A. Boehner throughout his tenure.

"If you're not careful, by Christmas you resemble John Boehner," Gingrich warned.

Rep. Jason Chaffetz (R-Utah), who is running for speaker, said Ryan "checks every box."

"I think he'd be a wonderful speaker," Chaffetz said on ABC's "This Week with George Stephanopoulos."