The ride-hailing company's board is planning to vote on the choice in the next few hours, the people said.
But in his response, Kalanick claimed that Benchmark "began secretly planning an effort to oust him" and "executed its plan at the most shameful of times" following the death of his mother in a May accident.
In June Mr Kalanick told employees he needed time off to grieve for his mother, who died in a May boating accident.
Since 2005, Khosrowshahi has led the travel-booking site Expedia to become "one of the largest online travel companies in the world", according to the company. He also must deal with driver discontent, although Uber already has started to fix that by allowing riders to tip drivers through its app.
Along with some investors questioning Immelt's previous tenure at GE, turmoil on Uber's board reportedly played a major role in Immelt's decision to remove himself from the search.
Whitman tweeted last month that she had no intention of becoming the CEO of Uber, but by this weekend, she was clearly in contention.
Whichever seasoned executive will ultimately take the helm, Uber faces a wholesale crisis that will require a leader who can transform the company's culture while shoring up its business to go public, analysts say.
Whitman was on the short list for the CEO job but word is that she wouldn't take it if Kalanick was to remain active in running the company.
His leave was also touted as an act of valour, aimed at giving Uber time to rebuild its reputation after a long chain of scandals.
Kalanick was the longtime CEO at the ride-sharing service, but stepped down earlier this year after a string of legal and internal controversies under his leadership forced his resignation. If that were to happen, the VC firm would have the upper hand, and Whitman could be back in the running.
Uber has also faced an executive exodus this year.
The next chief will have plenty to do.
Uber engineer Susan Fowler alleged that she and other female employees at the company were sexually harassed.