Keane is currently manager of Maccabi Tel Aviv with Delap as his assistant.
talkSPORT understands that Keane and Delap were hidden in a panic room before being evacuated from the country.
It is understood that they have now fled to Greece.
Former Leeds and Tottenham striker Keane accepted the job in Tel Aviv in June, signing a two-year contract in his first managerial job.
Prior to that, he had spent a brief spell as part of Sam Allardyce's backroom staff at Leeds as they were relegated from the Premier League last season.
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Keane scored more than 400 goals for his previous clubs including Tottenham, Leeds, Liverpool, Coventry, West Ham and Aston Villa.
The 43-year-old is also the most-capped player and all-time record goal scorer for the Republic of Ireland.
His former teammate, Delap, was named as his assistant manager at the club.
Maccabi Tel Aviv are currently top of the Israeli Premier League, having won four and drawn one of their opening games.
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They are also competing in the Europa Conference League this season.