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As the top seed, Serena Williams cast a long shadow as she routed Madison Keys to advance.Credit Ben Solomon for The New York Times

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At age 35, Venus Williams has been making a gallant effort to win an eighth major tennis title, her last one coming seven years ago at Wimbledon. But if she is to do it at this year’s United States Open, it will come at a heavy cost. She will have to deny her younger sister the rare opportunity to win a Grand Slam.

Williams dispatched Anett Kontaveit, 6-2, 6-1, and will play Serena Williams, the No. 1 seed on a path toward history, in the quarterfinals of the Open on Tuesday night.

That familial matchup was confirmed when Serena Williams followed her older sister on court at Arthur Ashe Stadium and pummeled Madison Keys, 6-3, 6-3.

Both women played clean, dominating tennis on Sunday, and so for the 27th time as professionals — not including other instances in practices and informal hitting sessions — they will put their unique and sometimes uneasy sibling relationship on public display.

“I’m playing, for me, the best player in the tournament,” Williams said.

That was Serena Williams, not Venus, speaking.

But it is Serena who is the best player in the world. She is the defending Open champion and earlier this year won the Australian Open, the French Open and Wimbledon. She is now only three victories from winning the Grand Slam.

But Venus is three away from her first U.S. Open title since 2001 and is playing superb tennis as she enters the latest showdown with Serena in a lifetime of them. This one, though, could derail her sister’s date with glory.

“I don’t think anyone wants to be a spoiler,” Venus said. “I think people love to see history being made. No one is out to be a spoiler, but at the same time, you are focused on winning your match, even though the circumstances are really much different.”

Serena holds a 15-11 advantage against her older sister and has won six of their last seven matches. But Venus won the last match on hardcourts, a compelling three-set affair last year in Montreal. At the Open, they are 2-2, with Serena winning the last encounter with two tiebreakers in the 2008 quarterfinals.

2015 U.S. Open

There was a period from the 2002 French Open until the 2003 Wimbledon championships when they met strictly in the finals of major tournaments, and Serena went 5-0. Venus came back and beat Serena in the 2008 Wimbledon final in straight sets, and the next year, Serena rallied to vanquish Venus for that Wimbledon championship, 7-6 (3), 6-2. It was the last time they met in a major final.

They first played against each other professionally at the Australian Open in 1998, a Venus victory, and their most recent encounter was a typically uneventful straight-set 6-4, 6-3 win for Serena in the fourth round at Wimbledon in July.

There have been times when their matches seemed like a torturous exercise — not just for the sisters, but for the fans and other family members. When their recent Wimbledon matchup was set, their mother, Oracene Price, said it would be a good opportunity to visit Big Ben rather than sit through another of her daughters’ competitive encounters.

But lately, the sisters seem more at ease with the process, at least in discussing it.

“I think it is more fun than it used to be,” Serena said. “We really relish the opportunity. We’re both happy to still be involved in getting so far. It’s still super intense. She’s doing well and she wants to win this. So do I. It’s not easy.”

Both players enter the quarterfinals on waves of momentum. Serena erased some of the serving difficulties that plagued her in her previous two matches and never allowed Keys a break point. She won 80 percent of the points on her second serve in the first set, and committed just six unforced errors over all.

Venus, too, has looked dominant in her past two matches. Serena pointed out that no one has beaten her more often than Venus, and that she could hold the key to upsetting her.

And she does upset her, sometimes.

At Wimbledon, Serena conceded in a light moment that Venus’s dog annoys her. On Sunday, Venus was asked if she harbored any similar sibling resentments against Serena.

“Well, we have a family gathering every year, and every year I don’t get much say,” Venus said. “She always picks the theme, so that bothers me.”

But in their matches, there is nothing visible but mutual respect and support, and the sisters insist that when they are away from the court, they do not discuss tennis.

Even if it can be a little awkward at times, and even if Venus poses a challenge, Serena pointed out that there were good aspects to playing her sister. Mainly, if she cannot win a Grand Slam, at least Venus will have a chance to win another major tournament to complete a remarkable comeback.

“I feel like that’s what we always wanted, growing up,” Venus said. “Just to be out there on the big stage, duking it out when someone named Williams will win. That’s a given on that one.”