Police divers found a skull Sunday in the waters off College Point, Queens.
The Oquendo family is holding on to hope while they wait for the test results, which are expected by Wednesday.
Over the weekend, a second arm and teeth were recovered from the river, along with clothes matching those worn by Avonte.
The 14-year-old was last seen on video tape walking out of his school in Long Island City in October.
Previous remains, said to be a human arm, legs, ribs and a pelvic bone, were found on a rock in the water near the College Point Yacht Club around 7 p.m. Thursday, about 11 miles from where Avonte was last seen.
Police responded to a 911 call in the vicinity of Powell Cove Boulevard and Endeavor Place after a 14-year-old girl stumbled across the limbs.
The remains were taken to a morgue, where the medical examiner's office will try to identify them and determine a cause of death.
A pair of sneakers and jeans were found that matched what Avonte when he disappeared. They are the same sizes that Avonte wore.
The family's attorney, David Perecman, says DNA tests will provide the final say and that relatives are not counting on anything until that determination has been made, however hope is quickly fading.
"It's not Avonte until the DNA evidence comes in and says it is," Perecman said. "On the other hand, it's not looking great. Maybe it's possible it will be some sort of closure."
Avonte's disappearance sparked a citywide search that included hundreds of officers, marine units and volunteers.
Missing person posters were plastered on lampposts and placed on car windshields throughout the city. The teen, who did not speak, was fascinated with the subway system and Metropolitan Transportation Authority officials made announcements on trains for weeks asking for help finding him. Police checked every subway station and tunnel.
Mayor Bill de Blasio said his prayers are with the family.
"There was an extraordinary mobilization all over this city by community members, by the NYPD, looking for this young man with no success," de Blasio said. "And I can only imagine what his parents have gone through."
If the remains are those of Avonte, sources say one possibility is that he may have fallen into the river next to his school.