"The date for [our surrogate] to deliver is the 31st," Shepherd's estranged husband Lamar Sally told PEOPLE. "[She's due] any time this week, actually. I am on baby watch."
Sally, who was married to Shepherd for nearly three years, filed for legal separation on May 2.
According to Los Angeles Superior Court documents, he has asked for full legal and physical custody of their unborn son.
However, no agreement has been reached between Shepherd, 47, and Sally, 43, regarding the child's custody.
"It's still unclear what's going on," said Sally.
Sally married the former View co-host in Chicago in August 2011.
In addition to asking for custody of the baby in his separation papers, Sally requested that Shepherd only have "reasonable" visitation rights. He also asked for an order preventing her from interfering with the baby being given to him after he's born.
He asked for spousal support and for Shepherd to pay his attorney fees. Sally also asked that the couple's prenup be invalidated based on fraud.
Shepherd, who had struggled to get pregnant in the past, has a son, Jeffrey, 9, with ex-husband Jeff Tarpley.
She and Tarpley split in 2010 after nine years together.