Two lanes are currently blocked at junction 4 at Purbrook after the lorry, which was travelling northbound, crashed into the central reservation and ended up in on the southbound carriageway.
Police confirmed that no other vehicles were involved and no injuries had been reported.
Officers were called just before 11.30am today.
For the latest travel updates following the incident check our live blog.
They have confirmed the driver of the HGV, a 22-year-old from Havant, provided a positive drug wipe at the scene and was arrested.
He has been released under investigation, and enquiries are ongoing.
Hampshire’s roads police team tweeted: ‘Driver of HGV involved in the collision on the A3(M) has been arrested and taken to Police Custody for providing a positive drug wipe #23754’.
The second lane on the northbound and southbound routes are shut to traffic. Police have said the closure could last ‘for some time’.
South Central Ambulance Service said it was not called out to the incident.
There are northbound queues going as far back as the A27 junction at Farlington.