Mr Elvidge, the firm's London general manager, told the Sunday Times: "While we haven't been asked to make any changes, we'd like to know what we can do."
He added: "That requires a dialogue we sadly haven't been able to have."
A TfL spokesman said it would be inappropriate to comment further.
Uber has been asking to meet Sadiq Khan, the mayor of London, since his election in May 2016 but those requests have been rejected, according to sources close to the US company.
More than 680,000 people have signed an online petition in a bid to keep Uber operating in London after its licence expires on 30 September.
It has 21 days to appeal TfL's decision and can continue to provide its services in the capital "until the appeals process has been exhausted" - a point Mr Khan reiterated in a response to the change.org petition.
Despite the more conciliatory language from Mr Elvidge, Uber said in a series of tweets on Sunday that it would challenge the TfL decision "in the courts to defend the livelihoods of drivers & the consumer choice of millions of Londoners who use Uber".
"Drivers who use Uber in London are licensed by TfL and have been through the same enhanced DBS [Disclosure and Barring Service] background checks as black cab drivers.
"We have always followed TfL rules on reporting serious incidents, with a dedicated team that works closely with the Metropolitan Police."
However, criticism of Uber continued on Sunday, with shadow chancellor John McDonnell calling the firm "a disgrace".
He told ITV's Peston on Sunday: "Hand on heart, I don't think I've ever used Uber. You have to abide by the law. If the company was outside the law, what could Transport for London do?"
Unite union boss Len McCluskey said it was part of a "horrible, race-to-the-bottom culture", while TUC general secretary Frances O'Grady called Uber a "global bully boy".
But Mr Hands said it was "astonishing Labour find it so easy to condemn Uber, a company with 3.5 million users".
"Of course Uber need to meet safety standards, but Labour always take it too far and simply don't have the balanced approach the country needs at this time," he added.
Although there are apps such as Mytaxi and Gett that allow passengers to hail black cabs in London, as well as those from minicab firms such as Addison Lee, Uber has no direct rival in the capital.
Estonian-based Taxify was forced to suspend services just days after launching in London earlier this month because it did not have a TfL licence.
The situation is different in the US, however, where Lyft has won market share this year following a series of PR disasters by Uber that resulted in the ousting of founder Travis Kalanick as chief executive in June.
Uber's share of the US ride-hailing market has fallen from 91% in 2014 to 74.3% in August, according to data from Second Measure, which tracks credit card purchases.
Lyft has 23.4% of the market, with other apps such as Via, Juno, Gett and Sidecar on 2.2%.
The Sunday Telegraph reported that senior Lyft executives had three face-to-face meetings with TfL and City Hall officials in the past year, as well as two conference calls.
The meetings, revealed in a Freedom of Information request, do not indicate that Lyft is preparing to launch in London, however.