Charlton have been given permission to speak to Northampton boss Chris Wilder and his assistant Alan Knill after triggering a clause in their contracts.
The Cobblers finished this season as League Two champions under Wilder.
“My relationship with the chairman and the fans is going to make this a very difficult decision, but I do feel at this stage of my career it is one I need to explore,” said Wilder, 48.
Jose Riga resigned as Addicks boss after their relegation to League One.
The Belgian had returned for his second spell at The Valley in January following the sacking of Karel Fraeye, but could not prevent the club from dropping into the third tier.
Charlton fans held a series of protests on Saturday as the Addicks lost 3-0 at home to Championship winners Burnley, the latest in a series of demonstrations against owner Roland Duchatelet.
Under Wilder, Northampton finished 13 points clear of second-placed Oxford in the League Two table after a 24-game unbeaten run.
Their promotion was achieved despite financial uncertainty earlier in the season because of outstanding loan repayments of £10.25m to the local council and a winding-up petition from HM Revenue & Customs.
BBC Radio Northampton are hosting a Monday Night Sport phone-in from 18:00-19:00 BST to discuss the possible departure of Wilder.