"We would like them to release her from jail and provide reasonable, sensible accommodation so she can do her job," one of her lawyers, Horatio Mihet, said in a statement. "That would be taking her name off of marriage licenses in Rowan County and allowing her deputies to issue the licenses."
Kentucky Gov. Steve Beshear's office said Monday he won't respond, noting that the conflict was a "matter between her and the courts."
On Sunday, her lawyers appealed the contempt of court motion that sent her to jail five days ago. Her lawyers said they'd file arguments to back up their appeal later Monday or Tuesday.
Davis was reading the Bible in jail and "exudes gentleness and peace," Mihet said.
"Her spirits remain high," he said. "She was brought to tears when she heard that so many people outside the jail and around the country are praying for her."
Supporters of Davis rallied Monday outside the home of U.S. District Court Judge David Bunning, who found Davis in contempt of court, CNN affiliate WCPO reported
, and also outside the jail where she's being held, CNN affiliate WSAZ reported.
Davis had refused to give licenses
to same-sex couples after June's Supreme Court decision on grounds that issuing the licenses would violate her Christian convictions against same-sex marriage.
Bunning ordered her to jail Thursday, ruling she was in contempt of court for refusing to issue the licenses and not allowing her deputies to distribute them for her. He said Davis would remain behind bars until she complies.
Five of her deputies agreed Thursday to issue marriage licenses in her absence, and the Rowan County Clerk's Office began doing so the following day.
How long will Davis stay in jail?
Mihet said on CNN that Davis was "willing to stay in this jail as long as it takes in order for her to win back her constitutional rights not just for her but for Americans of all faiths."
On Tuesday afternoon, a rally in support of Davis is scheduled to be held outside the Carter County Jail =m where she's being held. Mike Huckabee, a Republican candidate for president,
said he plans to visit Davis in jail before the rally.
Because of Huckabee's visit, five schools in east Carter County will close on Tuesday to alleviate traffic, said Sgt. Travis Steele with the City of Grayson. The closings will affect more than a thousand students, Steele said.
One of Davis's lawyers, Mat Staver, founder and chariman of Liberty Counsel, said Beshear could issue an executive order to solve the problem.
The legislature could pass a law removing clerks' names from the licenses, but it won't be in session until January.
But Beshear said last week he won't call lawmakers for a special session to deal with the issue, adding that to do so would cost "hundreds of thousands of dollars of taxpayers' money."
The governor added that he has no power to remove Davis from office.
In court papers, attorneys for Davis argued that she is unable to comply with the court orders because issuing same-sex marriage
licenses "irreparably and irreversibly violates her conscience."
A federal prosecutor said it was time for Davis and her county to comply with the judge's ruling.
"Government officials are free to disagree with the law, but not disobey it," U.S. Attorney Kerry B. Harvey said in a statement. "The county clerk has presented her position through the federal court system, all of the way to the U.S. Supreme Court. It is time for the clerk and the county to follow the law."
Supporters say Davis should not have to resign
Davis' lawyers said they were surprised when the judge ordered Davis jailed. They expected some other legal action, such as a fine.
Daniel Canon, an attorney who was working with the ACLU on the case against Davis, said his clients had not asked for Davis to be jailed. But now that she is, he said, there should be "some assurance that Ms. Davis is not going to continue to impose her religious beliefs."
Ryan Anderson, a senior research fellow at the Heritage Foundation, a Washington-based conservative think tank, said the legislature should remove clerks' names from the licenses as Davis has asked.
"Hopefully she'll get out of jail because the state of Kentucky will realize that there are compromises we can reach that will protect both the rights of gays and lesbians to receive marriage licenses and the rights of someone like Kim Davis not to have her name on that marriage license," he said.
Staver said Davis does not plan to resign. Anderson acknowledged that Davis could resign, but he said she shouldn't have to.
"We have a rich history in the United States of accommodating conscientious objectors," Anderson said. "Kentucky accommodates conscientious objectors for other types of licensings. ... The question should be: If we can accommodate someone, why shouldn't we?"