Family and friends of the 28-year-old appealed for the public’s help earlier this month and started a crowdfunding page online.
A total of 125 people have donated to the page, which as of midday today had raised £3,778.
They are aiming to raise £6,000.
Mr Bessey went missing from his home in Waterlooville on July 9, and his body was found in woodland on the outskirts of the town three days later.
People who have donated to help with funeral costs also left messages of support for the family.
Pat Ayrton commented: ‘RIP Anthony. Sympathy to all his family at this sad time.’
Luke Quinell said: ‘So sorry for your loss, our thoughts are with the whole family. xx’
And Stacey Partington-Williams posted: ‘I’m so sorry for your loss, my thoughts and prayers are with you xx’
A post on the page, from Anthony’s niece Toni Micklewright, said: ‘Ant was a beautiful human with a heart of gold, this awful news has shattered our hearts.
‘He put everyone before himself, and was the life and soul of any party.
‘We are raising as much money as possible to give Anthony the biggest and most perfect send off as possible. This is the least that he deserves.’
To donate to the JustGiving page click here.