The bill, which was heard in the House Judiciary Committee on Wednesday morning, has bipartisan support but faces opposition from those who view it as an attack on parental rights.
The main proponents of the bill are grandparents who have been cut off from their grandchildren.
Ruth Wade, 69, of Lacey, said her greatest regret is that she has not been allowed into the life of her great-granddaughter.
"I have missed out on all of her firsts, and unless this bill is passed I feel I might miss out on her first kiss, her first date, her fist high school prom and all these things," Wade said.
Opponents of the measure said that parents should have the right to decide what is best for their children unless the state determines that they are unfit to do so.
"The Legislature cannot mediate every family feud and every argument that adults end up in that affects children," said Joseph Backholm, executive director of the socially conservative Family Policy Institute of Washington. "There has to be a limit."
House Bill 1506 would allow courts to grant visitation to a third party who can demonstrate an ongoing and substantial relationship with a child that if severed would likely lead to the child being harmed.
In 2000, the U.S. Supreme Court ruled that Washington state's laws granting visitation rights to third parties when found to be in the child's best interest infringed on the fundamental liberty of parents.
Five years later, the Washington Supreme Court ruled that a third party - typically an estranged same-sex partner of a biological or adoptive parent - who can demonstrate a permanent and unequivocal parent-child relationship can be granted the status of "de facto parent."
Under current law, parents must be deemed unfit before other third parties can get visitation against the parents' will.