Authorities said they believe Hauser and her 13-year-old son, Daniel, may be in Mexico -- or trying to get there -- to seek alternative treatments for the teen who suffers from Hodgkin's lymphoma.
"We're looking for a hopeful resolution, it's all about Daniel," said Jason Seidl, chief deputy at the Brown County Sheriff's office.
Seidl declined to comment on whether Colleen and Daniel had been spotted and again urged anyone who thinks they may have information that could lead to their discovery to immediately contact the police.
Late Thursday afternoon, Anthony Hauser made an emotional appeal to his wife to return home with their sick son.
"This is for you Colleen if you're out there," Hauser said from his farm in Sleepy Eye, Minn., to his missing wife. "Please bring Danny home so we can decide as a family what Danny's treatment should be.
"I know you're scared and I feel that you left out of fear, maybe without thinking it all the way through," said Hauser.
"Danny is my son and the rest of the family is worried sick about how he is," Hauser said. "Please call me and let's talk about how we can work this out."
Hauser repeated, "Please give me a call."
Brown County Sheriff Rich Hoffman also spoke at Thursday's press conference, pleading with Colleen to return to Minnesota.
"We're all working to the same end, to make sure Danny is back home and safe," said Hoffman. "I'm asking you to contact us to arrange your safe return."
"We will not take an enforcement action if you've shown a good faith effort to come back," said Hoffman.
"We are concerned other individuals are going to use this incident as a mean to forward their own agendas which might not be in Danny's best interest," he said.
Colleen and Daniel Hauser were last spotted in Southern California Tuesday morning, according to the Brown County Sheriff's office, who said it was "reliable information" that has led them to believe the duo is heading to Mexico to seek alternative cancer treatment.
Colleen Hauser and Daniel disappeared after a court rejected the boy's request to refuse chemotherapy treatment for his Hodgkin's lymphoma disease. Doctors believe Daniel will die without the treatment.
The Hausers have said that they would prefer a less rigid chemotherapy treatment combined with other alternative treatments.
The family is Roman Catholic and believes in the "do no harm" philosophy of the Nemenhah Band, a Missouri-based religious group that believes in natural healing methods.
The warrant for deprivation of parental rights will ensure extradition to Minnesota if she is apprehended, according to police.
Authorities said the fugitives do not own a car and don't believe they could have left Sleepy Eye without help. They say they are following "any and all leads" and have been receiving tips from across the country.
Hoffman said that authorities do not have a specific vehicle they are looking for.
Colleen Hauser's husband, Anthony, said that he believes his wife saw X-rays of Daniel that made her scared and prompted her to flee, missing a court appearance Tuesday.
Hauser has been cooperating with police, but Hoffman declined to speculate on the "sincerity of the information."
Daniel's name has been added to the database of the national Missing and Exploited Children's Network, said Hoffman, who hopes the move will help spot the teen if he attempts to cross a border.
The Associated Press contributed to this report.