Ireland's Porter, 25, suffered a toe injury as Leinster lost to Glasgow in the Pro14 Rainbow Cup on Friday.
Sinckler, who toured New Zealand with the Lions in 2017, will join the squad at the end of Bristol Bears' season.
The prop, 28, gave an emotional television interview in May after not being selected for the Lions.
Speaking after his man-of-the-match Premiership display for Bristol against local rivals Bath, Sinckler said his omission from the squad "had been tough" because it "meant so much".
