Wales stand on the verge of securing last-eight status for just a fourth time in seven World Cup tournaments.
For such a scenario not to unfold, Samoa would need victory over world champions South Africa in Auckland on Friday, leaving Fiji then requiring a repeat of their 2007 World Cup win against Wales.
Williams is among the survivors from that 38-34 defeat in Nantes. He scored one of his team's five tries, but was powerless to stop Fiji prevailing as Wales went home painfully early and coach Gareth Jenkins lost his job.
Waikato Stadium is the venue for Sunday's reunion, although Williams' potential involvement is unclear as he recovers from a thigh muscle strain.
Wales coach Warren Gatland will announce his team overnight, but it would be no surprise if Williams was held back for Ireland in Wellington on Saturday week.
"You want to be involved in as many games as possible," he said.
"I have been chomping at the bit to play in this World Cup.
"But it is an injury that could have kept me out for longer, so I have got to count my blessings there, really."
Williams' team-mates against Fiji in 2007 included current World Cup squad colleagues James Hook, Stephen Jones, Mike Phillips, Gethin Jenkins and Alun-Wyn Jones, and he recalled: "That is not the way you want to be going home from the World Cup.
"It is sure to be in the back of some of the guys' minds. But to be fair, we haven't really mentioned it in this World Cup. We didn't want to jinx ourselves.
"We are looking really sharp, and I am sure if we can play our style of rugby and not be drafted into the way Fiji want to play, we will win on Sunday.
"This is probably the most mentally tough group I've ever been involved in.
"It has taken a while. We have always been a very good rugby side, but sometimes we've lacked the mental toughness out there. We are in a good place at the moment.
"We know the mathematics if things go our way and what it could mean, but as far as we are concerned it is about performing on Sunday and winning that game.
"Regardless of the South Africa versus Samoa result on Friday, it's a must-win game for us on Sunday. You don't want to go into a quarter-final having lost your previous game."
Fiji's form has been underwhelming in the tournament so far. Although they beat Namibia easily, they then scored just 10 points and conceded 76 in being despatched by South Africa and Samoa.
But Williams said: "It is a very good Fiji side that can play some rugby if you let them. We need a big performance.
"It has been a tough group. Fiji had two tough games against a very good South African side and Samoa, who I feel are a great side.
"It hasn't been easy for Fiji, and hopefully it won't get any easier on Sunday.
"We have got complete respect for them. You can't not respect a team that has beaten you and had a draw against you over the last few years.
"When they start playing the rugby they want to play, there are not many teams that can compete with them. They are a very dangerous side.
"They probably haven't played the expansive rugby that they are used to playing, but the teams they've played against have stopped them doing that, and it is up to us to do exactly the same.
"We are a proud bunch of guys, and as far as we are concerned we want to go out and play to the best of our ability and beat Fiji on Sunday."
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