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".
"I am not going to lie - I am emotional," Sinckler told BT Sport
"I'm lucky I had my mentor and we broke it down. I will understand it and in a year's time it will all make sense, even if it doesn't right now.
"I have never experienced something like this in my whole life."
Sinckler's interview won praise from former players and pundits alike, and Lions head coach Warren Gatland said he has been "very pleased" with the prop's response to being left out.
"It's really bad luck for Andrew and we send him our very best wishes for a quick and successful rehab," said Gatland.
"Andrew is a young man with a big future ahead of him and I'm sure he'll be in contention again in four years' time.
"As always in rugby, when there's an injury, there's an opportunity for someone else. I have been very pleased to see the way Kyle has responded to initially being left out of the touring party.
"He's showed some excellent form of late and I hope he continues to stick two fingers up to me about leaving him out in the first place."
The first Test against the Springboks will take place on 24 July in Cape Town (17:00 BST).