Police said the man was detained as part of an ongoing investigation into the conflict in Syria.
Officers from the South East Counter Terrorism Unit (SECTU) and Sussex Police are searching one residential address.
The arrest is not linked to any immediate threat in the UK, police said.
Crawley MP Henry Smith said police told him early this morning that a 40-year-old man had been arrested at a property thought to be in the Langley Green area of the town.
He said: "I understand he has been arrested in relation to membership of organisations which are connected with activity to the Syrian civil war."
Mr Smith said he understood people would be concerned over news of the arrest, particularly following the death of Abdul Waheed Majeed, from Crawley, who died in a suspected suicide attack in Syria in February last year.
The 41-year-old, who was born and raised in Crawley, drove a truck bomb into the gates of a prison in Aleppo.
The father of three had left Britain in 2013 telling his family he was going on a humanitarian mission to Syria.
Mr Smith said: "These cases are in the minority but they are very concerning and the public need to feel that they are as protected as they can be."
A spokesperson for SECTU said the unit was committed to intervening in activities "which cause concern to local communities".
"We will continue to work with our partners and local communities in order to protect them from individuals and organisations which may seek to exploit them for extremist purposes," the spokesperson added.
Local residents are being asked to contact police if they have any concerns regarding extremist activity in the area.
A 40-year-old man has been arrested in Crawley, West Sussex, on suspicion of terror offences.