Gabriel Johnson hasn't been seen since December 26, police said, and his mother was arrested last week in Miami Beach, Florida, after not reporting for a December 28 child custody hearing in Arizona.
"We have some good indications at this point with our investigation that Tammi and Jack Smith do know more information than they have provided to us that could possibly lead us to Gabriel," Sgt. Steve Carbajal, spokesman for the Tempe (Arizona) Police Department, told HLN's Nancy Grace on Thursday.
For their part, the Smiths told HLN's Jean Casarez on "Nancy Grace" that they voluntarily took polygraph tests Friday at the Tempe Police Department.
"The polygraph test is not like what you see on TV," Jack Smith said. "It's almost like being strapped into the electric chair, so it's very intensive. And we were very happy to do it."
The Smiths have said they met Elizabeth Johnson, 23, seven months ago during a long layover at an airport, and later the three discussed a plan to adopt Johnson's child, Gabriel.
But the boy's father, Logan McQueary, has legal custody and has said Elizabeth Johnson urged him to sign papers giving the Smiths custody of the boy, but he refused.
"She didn't want Logan to have the baby, and we couldn't adopt the baby because Logan wouldn't sign the papers," Tammi Smith said on "Nancy Grace."
"So her idea was to just keep running forever. And we told her, 'You can't run forever. They're going to find you.' "
Elizabeth Johnson has refused to say where the boy is, according to police, and remains in a Florida jail on suspicion of custodial interference.
Carbajal's department has indications that Gabriel is alive, he said, adding that the Smiths have spoken with investigators. But he declined to say why police believe that the couple has more information.
The Smiths say they do not know where Gabriel is.
Detectives have not been able to confirm that Johnson gave the child to an unidentified couple in San Antonio, Texas, as she has claimed.
"On one hand, we hear that that went down and the child was given away to an unknown couple at a park," Carbajal said.
"We balance that with statements made by Elizabeth that she killed Gabriel. Which story is true? Our investigators are trying to get to the bottom of that right now."
New leads and developments in the case were coming in by the minute, he said. The FBI found Johnson's car in San Antonio and have examined it for any clues that could lead them to the boy, but his whereabouts remain unknown.
Anyone with more information is asked to call the Tempe Police Department at 480-350-8311 or the National Center for Missing & Exploited Children at 1-800-THE-LOST.