For all the impatience in the game for the next generation to start grabbing the biggest prizes, the likelihood is that the established powers will prevail here again this year, an inkling given strength on day two of the French Open by the defending champion, Novak Djokovic, and Rafael Nadal, who stands on the threshold of more French history.

If the Spaniard can build on his encouraging start – a 6-1, 6-4, 6-1 win against the French world No45, Benoît Paire, on Court Suzanne Lenglen in 1hr 52min – he is headed for a semi-final collision with Djokovic. The Serb’s campaign began in similarly ominous fashion as he took just under two and a half hours to win 6-3, 6-4, 6-2 on the showpiece court, Philippe Chatrier, against Nadal’s compatriot Marcel Granollers, ranked 77.

Djokovic’s first match with Andre Agassi as coach could hardly have gone better, although he confirmed later it would be a short first stint. “He’s going to stay I think – I hope – until the end of this week. Then he has to leave because he has some things that he cannot reschedule.”

However, Djokovic said, it has already been productive. “So far we have plenty of information, plenty of things to process. This is exactly what I need at this moment, a person like him [who] understands the transitions as a tennis player and as a person, as well, going through lifestyle and certain choices that you make, how that affects you later on. He has been through that and he has a lot to share with me outside and inside of the tennis lines. I’m really enjoying it so far.”

Nadal remains the bookmakers’ choice to win his 10th title here, something no one on the men’s or women’s tours has done at any major in the open era, which began in 1968. Margaret Court (not everyone’s favourite Australian after an outburst against same-sex marriage last week) won 11 Australian titles straddling the transformation period but in the early days the tournament often did not attract the world’s best players.

The Spaniard warmed up for the La Décima challenge by getting to double figures in two of his other favourite clay tournaments, Barcelona and Monte Carlo, as well as his fifth in Madrid. Reaching for an eighth Italian Open title, he suffered his only clay setback in 18 matches to that point, against the rising Dominic Thiem.

Aljaz Bedene is also having a storming season on clay – albeit at a lower level – and won his 21st of 24 matches on the surface to reach the second round with an impressive 6-4, 6-0, 3-6, 6-1 victory against the hard-hitting American Ryan Harrison, who is 10 places ahead of him in the world rankings at 42.

Bedene confirmed later he is splitting his time between Welwyn Garden City and his birthplace of Slovenia, and has set himself a goal of playing in the 2020 Olympics in Japan – although, given his failed efforts to represent Great Britain in the Davis Cup, he was honest about his divided loyalties.

“At the moment for the clay season I spent quite a lot of time in Slovenia. My fiancée wants to have her career there. I’m travelling a lot, so it’s not easy, but I do want to spend as much time as I can with her. I’ve still got a house [in Welwyn]. My home is still there.”

Pressed on comments by Dan Evans that he is not properly British, Bedene said: “At the moment I’m representing Great Britain but I do want to play Olympic Games. That’s a dream for every athlete. I don’t know what the rules are, really.” He would at least have to make himself available for his home federation but therein lies a legal minefield. For now, he said: “I am playing for myself. I’m 27 years old. I want to have a good career.”

Garbine Muguruza is the reigning women’s French Open champion. Photograph: Gonzalo Fuentes/Reuters

Bedene played some sublime tennis in all but the third set and, even then, remained in touch with his opponent.

Harrison, who has never properly harnessed his talent, competed hard in the first 45 minutes but collapsed in a heap of frustration in the bagelled second, smashing his racket to the ground and arguing with the umpire as Bedene moved smoothly through the gears. Harrison broke a seven-game losing streak at the start of the third and his spirits rose as the match turned quickly from a mugging to a fair fight but the end was mercifully swift.

Bedene brought the match to an emphatic conclusion with his 12th ace and will need to serve like that against Jiri Vesely, another of the gathering youngsters knocking at the castle gate, who beat the American 14th seed, Jack Sock, 7-5, 7-5, 6-3.

“I think Vesely can beat many good players on clay,” he said. “I don’t think Sock is the fittest; Vesely is really fit. He’s a great player. I didn’t think Sock was going to win.”