The 27-year-old Senegal international is expected to complete a season-long loan move.
He still has three years remaining on his Toon deal but looks set to spend the campaign back in Turkey following a loan spell at Sivasspor during the first half of the year, say the Chronicle.
Saivet was snapped at an airport and the image was shared on social media.
Rafa Benitez's side have been actively looking for clubs to take Saivet.
They were looking to offload a high earner – on around £35,000 a week – who was deemed surplus to requirements.
The Chronicle add it is not yet clear whether Bursaspor will play Saivet's salary in full.
But they warn history could repeat itself in the sense that United have had to party subsidise loanees' wages before.
Saivet has made only one league appearance for Newcastle over the course of the past two seasons.
He told BeIN Sport Turkey: "I am very happy that I came to Bursaspor. It was very important for me to be transferred here for my career.
"I will be training with the team after the health check tomorrow. We worked together too, and hopefully everything will be fine."
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