Daniel Farke has been desperate to sign a goalscorer this summer and Piroe has been one of his top targets.
Swansea rejected several offers for the Dutchman earlier in the window and several clubs cooled their interest in him.
But Leeds continued to persist to try and sign Piroe given their desperation for a forward and a deal is now close to being completed.
A £12m fee has been agreed upon between Leeds and Swansea and the forward is now travelling to Yorkshire for a medical.
The Whites will put him through a routine medical before the striker signs a long-term contract.
Piroe has been a proven goalscorer in the Championship, netting 19 goals for Swansea in the previous campaign.
He has resisted offers of a new deal and Swansea boss Michael Duff admitted earlier today that he would have to be sold if he does not agree to sign an extension.
