Rodgers impressed in an interview with chairman Huw Jenkins on Wednesday evening and was offered the job soon after.
He is set to be unveiled by the weekend and could be in the stands for this evening's friendly fixture at WPL outfit Neath.
The 37 year old, who can speak Spanish and Italian, was sacked by Reading in December after a torrid run of results which left the Royals deep in the relegation zone.
At Watford, he enjoyed 12 wins in 31 matches and helped to keep the Hornets in the Championship before moving to the Madejski Stadium for £500k.
Swansea have been on the lookout for a new manager since Paulo Sousa left for Leicester City last week.