Lieberman may lose his committee chairmanship this week, as Senate Democrats meet on the issue.

(CNN) – Two members of the Senate’s Democratic caucus have called for Joe Lieberman to lose his committee chairmanship. On Sunday, North Dakota Sen. Byron Dorgan didn’t go that far — but did say the Connecticut senator’s actions during the campaign were “not acceptable,” as Republican Sen. Jon Kyl said the GOP would “welcome [Lieberman] with open arms.”

“…As a chairman of one of our significant committees in the Senate, not just going off and supporting a presidential candidate of the other side but also criticizing the candidate on our side, and also involving himself in a couple of Senate races on the other side,” Dorgan said on Fox News Sunday, when asked whether he thought the Democrat-turned-independent should lose the chairmanship of the Senate’s Homeland Security Committee.

“The question is, is that acceptable? The answer is no. The question, I think, mainly is what should happen as a result of that.”

The full Democratic caucus is meeting this week to vote on what penalty Lieberman may face for his decision to endorse John McCain’s presidential bid and criticize Obama. “We will hear from Sen. Lieberman, and then we’ll make a judgment,” said Dorgan. “I was concerned and was upset about what happened earlier this year. But you know what, we turn the page, we’ll make a decision about that on Tuesday.”

Arizona Sen. John Kyl, also appearing on Fox News Sunday, said the GOP would “welcome [Lieberman] with open arms… Joe is a very open-minded person. People in both caucuses love Joe because he is a great American. He’d be welcome in our conference, we’d love to have him there. And since he was of such great assistance to John McCain in this election, it would be good to have him with us.”

On Friday, both Vermont senators — Judiciary Committee Chairman Pat Leahy and Bernie Sanders, an independent who caucuses with Democrats — both said Lieberman’s actions during the campaign should cost him his chairmanship.