Haseeb and Christy Amireh, of San Jose, California, had a baby with the help of a surrogate mother in the Mexican city of Villahermosa, in the southern state of Tabasco.
Their son, Grayson, was born on April 16. The Amirehs were present at the birth, and everything to that point had gone according to plan.
But since Grayson's birth, Tabasco state authorities have not provided the parents with a birth certificate.
According to the couple, Mexican officials told them laws regulating surrogacy had changed and they could no longer get the documentation to return to the United States.
On Friday, the Amirehs said a resolution may be close, but declined to talk about specifics.
"The momentum to be able to go back home is quickly building," Haseeb Amireh said. "We may have good news as soon as this weekend."
The couple set up a fundraising website to help with legal costs.
"Due to politics the current administration has revoked our human right to receive a birth certificate for our child and no passport to fly home," the couple's Go Fund Me page
states. "This leaves us waiting in hotel rooms waiting without end in sight."
The Tabasco government did not comment on the claim that it was refusing to grant the documents, and referred CNN's calls to the state's Civil Registry Office.
The Amirehs "have submitted their adoption contract and it is currently being reviewed," said Maria del Carmen López, an official at the office.
She declined to comment further, citing privacy laws.
On her Facebook account, Christy Amireh posted pictures of her baby with the words "Bring Grayson Home!"
The case grabbed the attention of U.S. Rep. Eric Swalwell Jr., who represents the couple's district in California.
"From what we learned, the governor in that state has put a moratorium on all birth certificates for surrogate families. I think this highlights why it's really important to understand the laws of any country you are traveling to, to make a health care decision," Swalwell told CNN affiliate KPIX.
It was unclear what modifications to the surrogacy law might be keeping the American couple stuck in Mexico.
But lawmakers in Tabasco have called for regulations on surrogacy births, and the governor supports the move, according to local media reports.
The issue of surrogacy in Mexico is also getting attention at the federal level.
Last month, Mexico's Congress published a book that investigated alleged abuses and exploitation of surrogate mothers. Some federal lawmakers are pushing for a national law to regulate such services.
In recent years Tabasco has increasingly become a destination for American couples in need of surrogacy services, according to Marcy Darnovsky, executive director of the Center for Genetics and Society in Berkeley, California.
"The cost is a big reason. We're talking about people who are trying to arrange surrogacy that is significantly less expensive than it would be in the U.S. That's why people go," Darnovsky said.
In 1993, Tabasco state enacted a provision in the law that regulates surrogacy, one of a few Mexican states where this is allowed.
"Tabasco is closer than other countries that offer these services like India. There are companies in Mexico that serve as brokers who can put couples in touch with surrogate mothers. They say come to Cancun, have a holiday, and we will assist you with your surrogacy needs," Darnovsky said.