The device, described as a pipe bomb, was discovered on a pathway in Foxes Glen during follow-up searches relating to a gun attack on Thursday.
Police said it may have been thrown at three PSNI officers who were shot at as they attended a call-out in the area.
The army carried out a controlled explosion on the bomb amid a major security alert on Friday and it was taken away for further forensic examinations.
"Police inquiries into this attempted murder are active and ongoing," the PSNI said.
"We would appeal to anyone with information about activity in the wooded area beside Foxes Glen during the morning of Thursday 16 May to contact police on 0845 600 8000."
Up to six shots were fired during the attack on police. Witnesses say the officers dived for cover and crawled on the ground until they reached their own vehicle at around 1pm.
The female police officer and her two male colleagues were left badly shaken, but returned to work on Friday.
The incident happened in a residential area where children were playing nearby. Police added that the bomb was found on a footpath often used by residents.
Homes in Foxes Glen were evacuated during the security operation which ended at around 10.15pm on Friday evening. During previous searches, officers recovered a number of replica firearms and ammunition in Belfast.
A 26-year-old man was also arrested and questioned in police custody at Antrim Serious Crime suite. He was later released unconditionally by detectives.