Jose Mourinho's men claimed the trophy by beating Ajax 2-0 at the Friends Arena on Wednesday evening, becoming the fifth team to win all three major European trophies.
Despite this, United aren't planning a parade following the British government's decision to raise the national threat level from severe to critical in the wake of Monday's attack on Manchester.
An explosion at Manchester Arena killed 22 people and injured 59, with Chelsea subsequently cancelling their plans to celebrate their Premier League title.
However, Neville, who claimed six Premier League titles with United during his spell with the club, believes a parade would unite the people of Manchester.
"When we won the Champions League we paraded the trophy in the city centre, ending in the Arena and celebrated there. It was unbelievable," Neville told BBC Radio 5 live .
"After what happened on Monday, United should parade the trophy around Manchester and finish at the Arena.
"I think it would be special. You should fill the streets of Manchester -- Deansgate, St Peter's Square, St Ann's Square -- and get the trophy back.
"Football can do brilliant things so don't go off on holiday, get back to Manchester, get the trophy out and share the success with the city."