Derbyshire are currently placed at the fourth spot while Durham are sixth in the County Championship 2022 Division Two points table.
Holders Glamorgan will begin their title defence away against Derbyshire in the inaugural contest of the Royal London Cup 2022.
Northamptonshire will face Derbyshire in the T20 Blast 2022 North Group contest on Tuesday.
India will face Derbyshire and Northampton in warm-up games in July before taking on England in the full limited-overs tour.
Suranga Lakmal was involved in heated exchange with a fan after his arrival at the airport for Sri Lanka's poor show against India.