Newcastle owner Mike Ashley and chairman Chris Mort approached Elland Road officials about the prospect of taking Wise.
Leeds announced they have agreed a compensation deal with the Magpies for his services.
A club statement read: "We have agreed a compensation package with Newcastle United for the services of our manager Dennis Wise and he will part company with the club on Tuesday January 29 after completing the preparations for our game at Southend United that evening.
"Newcastle had approached us for permission to speak with Dennis on Friday and since then things have happened very quickly.
"We will always be grateful to Dennis for the way he has turned things around at the club on the playing side and he goes with our best wishes for the future.
"We are not prepared to comment on the inevitable speculation that will now follow about who will replace Dennis. That is not the way we do our business at this club.''