The Sweden international, who spent the second half of the season on loan from Roma, will arrive for a fee of around £3million.
Olsen made only one appearance during his loan spell but made a positive impression at Bodymoor Heath and is viewed as ideal competition for No.1 Emi Martinez.
Olsen, capped 57 times by his country, previously had loan spells in England with Everton and Sheffield United.
Speaking to the Swedish media while away on international duty earlier this week, he remained uncertain where his future might lie but admitted he had enjoyed his time at Villa Park.
He said: “I like it very much there. I feel incredibly good. It’s a fantastic club and I feel that I have developed a lot. We will see how it turns out.”
The 32-year-old will follow Philippe Coutinho, Boubacar Kamara and Diego Carlos through the door with Villa making a rapid start to what is expected to be a busy summer.
Boss Steven Gerrard would like to add another left-back, midfielder and striker to the ranks before the transfer window closes.
But recruiting another centre-back is not on the agenda and reports linking them with Burnley’s Ben Mee are thought to be wide of the mark.