Sinckler was one of the most high-profile omissions from Warren Gatland’s original 37-man selection for South Africa.
He gave an emotional interview in which he said he had “never had so much anger inside” him.
Gatland said he was delighted with Sinckler’s response to his initial exclusion and said he hoped the prop would continue to “stick two fingers up to him”.
Porter, 25, was injured playing for Leinster in the 15-12 defeat by Glasgow Warriors at Scotstoun on Friday.
Sinckler, a Lions tourist in New Zealand in 2017, will join up with the squad following the conclusion of Bristol Bears’ domestic season. Bristol have qualified for the Premiership play-offs.
The Lions assemble in Jersey on Sunday to begin preparations for the match against Japan at BT Murrayfield on 26 June, before flying to South Africa.
“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.”