The 34-year-old former England international has been training with the club for around a month after cancelling his contract with Kayserispor by mutual consent.
He appeared in pre-season at Turf Moor against Newcastle United and Cadiz, and also featured at Gawthorpe against Stockport and against Manchester United at Old Trafford, where he scored in a 3-1 defeat, as Sean Dyche assessed his fitness.
And Dyche has decided to add to his squad, which has only Dwight McNeil and Johann Berg Gudmundsson as recognised senior wide players.
Lennon initially joined Burnley from Everton in January 2018, helping the club qualify for the Europa League in a seventh-place finish.
He scored his first and only goal for the club in a 4-0 win at home to Bournemouth in September 2018, and left the club in June last year at the end of his contract after making 55 appearances.
Lennon is eligible to play in Wednesday night’s Carabao Cup tie at Newcastle United.