Thompson is Jackett's first Pompey signing
The 26-year-old has joined on a two-year deal after running down his contract at Swindon.
Thompson, who can play in the centre of defence, right-back or in midfield, was captain of the Robins team that was relegated from League One last season.
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Hide AdHe made 36 appearances for Swindon last term, scoring twice.
In total, Thompson made 157 appearances for the Wiltshire club after graduating through their academy system.
He said: ‘I wanted to sign for a club of this size.
‘It’s an exciting opportunity and one I’m looking forward to.
‘It was about the potential this club has, and, with the new owners it’s a great opportunity.’