The discovery comes hours after two men were detained when houses in Ayrshire were raided by police investigating previous parcel bomb attacks against Lennon, a republican group and two prominent fans.
The raids also came after Lennon, a Northern Irish Catholic, was attacked by a rival fan during a match against Hearts – a traditionally Protestant team – in Edinburgh on Wednesday night.
The man jumped over a barrier from a home supporters' stand and ran down the touchline before attempting to strike the Celtic manager as he stood in the technical area beside the pitch. A 26-year-old man is due to appear at Edinburgh sheriff court on Thursday.
The package containing the bullet was found at the Celtic Park stadium, in the Parkhead area of Glasgow, on Thursday morning, but police said they were not currently linking the package to the incendiary bomb investigations.
Officers would not confirm the package's exact contents, but it is understood that it contained at least one bullet.
Lennon has had bullets sent to him in the past, allegedly from opponents linked to loyalist paramilitaries in Northern Ireland.
A police spokeswoman said the bomb squad had not yet been called out to deal with the latest alert, and the stadium's staff had not been evacuated.
"There has been a suspicious package found this morning. Police are just at the scene and are carrying out inquiries to try to establish what it is," she said.
Strathclyde police raided a number of houses in the Kilwinning area just after dawn and held two men, aged 41 and 43, in a major operation over the posting of five viable incendiary devices in March and April.
The force said the two men detained in Kilwinning were held under the Explosive Substances Act 1883, and that the operation would continue.
Chief Superintendent Ruaraidh Nicolson, the force's head of community safety, said: "Naturally, given the significant police presence here, people are bound to be concerned.
"I would like to reassure everyone who lives here that they are not at risk and that we are fully in control of the situation.
"I would expect that there will continue to be a fair amount of activity in this area throughout the day. I appreciate that that may well be inconvenient to people who want to go about their daily business.
"I can assure you that we will be doing everything we can to keep disruption to a minimum and to get the local area back to normal as quickly as possible."
Alex Salmond, the Scottish first minister, said he had been briefed on the raids by police.
"I am very pleased and satisfied that the police are making substantial progress in this important investigation," he said. "I am sure that satisfaction will be shared by the Scottish public as a whole."
Lennon was the target of two devices, posted from the nearby town of Saltcoats in March. A third device was delivered to the office of Trish Godman MSP, a few days after she stood down as the deputy presiding office of the Scottish parliament.
A fourth parcel bomb was posted to Paul McBride QC, Lennon's prominent and outspoken lawyer, from a postbox in Kilwinning in April. It was the last of the devices to be sent during a seven-week period.
A week after the letter-bombing campaign came to public attention last month, it emerged that a fifth incendiary device had been sent to the Glasgow-based republican campaign group Cairde na hÉireann in March.