De Boer has been heavily linked with the now-vacant hot seat at Spurs, and has previously confirmed that he'd been approached by the north London club.
While Southampton manager Maurico Pochettino appears to be the slight favourite to take over at Tottenham following the departure of Tim Sherwood earlier this week, the Telegraph report that De Boer is still in the running for the job.
However the paper add that should De Boer get the nod then he won't take Bergkamp, currently his assistant at Ajax, with him because of his fear of flying that restricts his ability to attend European games away from home.
The news is likely to come as a relief for Tottenham fans given Bergkamp's connection with Arsenal. The striker made 423 appearances for the Gunners, scoring 120 times, and has been immortalised in a statue outside the Emirates Stadium.
No talks are likely to be held with De Boer until next week because he is currently away in Jakarta with Ajax - a trip which Bergkamp hasn't attended.
Southampton meanwhile have called a board meeting to discuss the future of Pochettino.
The south coast club are keen to retain his services but the Argentinian only has one year left on his current contract and is reportedly willing to talk to Tottenham.
Real Madrid boss Carlo Ancelotti remains on Spurs' radar.