MONACO — There was a time here, in this principality on the Riviera, when Dmitry Rybolovlev could have been mistaken for royalty as he traipsed between restaurants, trailed by up to five bodyguards, more than the monarch himself, Prince Albert II.

Some days the prince would join Mr. Rybolovlev at a luxury box in Louis II Stadium, where they cheered on Mr. Rybolovlev’s professional soccer team, AS Monaco FC, a source of fierce national pride. Afterward Mr. Rybolovlev would retire to his penthouse, the Belle Époque, with its kingly view of Port Hercules and the ancient palace, where he attended the christening reception for the prince’s twins in 2015.

Lately though, Mr. Rybolovlev, a billionaire Russian mining magnate, has been less visible and Prince Albert has maintained a delicate, diplomatic distance from him.

Mr. Rybolovlev is still a force here, of course, owing primarily to his soccer team, whose success drew a recent grateful acknowledgment from the prince.