Roman Abramovich hugged Jose Mourinho - just hours after being told Klinsmann did not want to become Chelsea boss.
It was hoped the much-heralded clinch after Sunday's FA Cup semi-final win was a signal their apparent feud had been forgotten, the Sun reports.
But a Chelsea insider revealed: "The meeting between them looked like an accident rather than something that had been planned by Roman.
"You wonder what Roman would have done if Klinsmann had said yes."
A delegation from Abramovich's bankers, Millhouse Capital, travelled to Los Angeles to talk to the German legend. But despite an offer of a potential £6.4million a year, Klinsmann told Chelsea he was too settled in America to uproot his family.
Klinsmann is thought to be an ideal candidate for the job, having already spent two successful spells in England, with Tottenham, during which he developed a rapport with both English fans and media alike and was voted Premiership player of the year.
Klinsmann, who resides in California, has been out of work since guiding Germany to the semi-finals of the World Cup last summer, but was thought to be close to taking over the US national team in December.
The 42-year-old was heavily criticised in his homeland for his preparations for the World Cup - his decision to live in the US in particular came under constant attack - but the Nationalmeinshaft's performance once there was hailed as a triumph, and the relatively inexperienced Klinsmann became something of a national hero, eurosport.com reports.
Prepared by Alexander Timoshik
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